Precalculus Examples

$x^{2} + y^{2} = 64$ , $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 1

Solve for $x$ in $x^{2} + y^{2} = 64$ .

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Step 1.1

Subtract $y^{2}$ from both sides of the equation.

$x^{2} = 64 - y^{2}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 1.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{64 - y^{2}}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 1.3

Simplify $\pm \sqrt{64 - y^{2}}$ .

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Step 1.3.1

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{8^{2} - y^{2}}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = y$ .

$x = \pm \sqrt{(8 + y) (8 - y)}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \pm \sqrt{(8 + y) (8 - y)}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 1.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 1.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 1.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{(8 + y) (8 - y)}$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 2

Solve the system $x = \sqrt{(8 + y) (8 - y)}, \frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{(8 + y) (8 - y)}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$ with $\sqrt{(8 + y) (8 - y)}$ .

$\frac{{(\sqrt{(8 + y) (8 - y)})}^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $\frac{{(\sqrt{(8 + y) (8 - y)})}^{2}}{16} + \frac{y^{2}}{9}$ .

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Step 2.1.2.1.1

Simplify the numerator.

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Step 2.1.2.1.1.1

Rewrite ${\sqrt{(8 + y) (8 - y)}}^{2}$ as $(8 + y) (8 - y)$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(8 + y) (8 - y)}$ as ${((8 + y) (8 - y))}^{\frac{1}{2}}$ .

$\frac{{({((8 + y) (8 - y))}^{\frac{1}{2}})}^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{((8 + y) (8 - y))}^{\frac{1}{2} \cdot 2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{{((8 + y) (8 - y))}^{\frac{2}{2}}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.1.4.1

Cancel the common factor.

$\frac{{((8 + y) (8 - y))}^{\frac{2}{2}}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$\frac{(8 + y) (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{(8 + y) (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.1.5

Simplify.

$\frac{(8 + y) (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{(8 + y) (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.2

Expand $(8 + y) (8 - y)$ using the FOIL Method.

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Step 2.1.2.1.1.2.1

Apply the distributive property.

$\frac{8 (8 - y) + y (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.2.2

Apply the distributive property.

$\frac{8 \cdot 8 + 8 (- y) + y (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.2.3

Apply the distributive property.

$\frac{8 \cdot 8 + 8 (- y) + y \cdot 8 + y (- y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{8 \cdot 8 + 8 (- y) + y \cdot 8 + y (- y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.3

Simplify and combine like terms.

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Step 2.1.2.1.1.3.1

Simplify each term.

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Step 2.1.2.1.1.3.1.1

Multiply $8$ by $8$ .

$\frac{64 + 8 (- y) + y \cdot 8 + y (- y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.3.1.2

Multiply $- 1$ by $8$ .

$\frac{64 - 8 y + y \cdot 8 + y (- y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.3.1.3

Move $8$ to the left of $y$ .

$\frac{64 - 8 y + 8 \cdot y + y (- y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{64 - 8 y + 8 y - y \cdot y}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.2.1.1.3.1.5.1

Move $y$ .

$\frac{64 - 8 y + 8 y - (y \cdot y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$\frac{64 - 8 y + 8 y - y^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{64 - 8 y + 8 y - y^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{64 - 8 y + 8 y - y^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.3.2

Add $- 8 y$ and $8 y$ .

$\frac{64 + 0 - y^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.3.3

Add $64$ and $0$ .

$\frac{64 - y^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{64 - y^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.4

Rewrite $64$ as $8^{2}$ .

$\frac{8^{2} - y^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = y$ .

$\frac{(8 + y) (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{(8 + y) (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.2

To write $\frac{(8 + y) (8 - y)}{16}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{(8 + y) (8 - y)}{16} \cdot \frac{9}{9} + \frac{y^{2}}{9} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.3

To write $\frac{y^{2}}{9}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{(8 + y) (8 - y)}{16} \cdot \frac{9}{9} + \frac{y^{2}}{9} \cdot \frac{16}{16} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.4

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.2.1.4.1

Multiply $\frac{(8 + y) (8 - y)}{16}$ by $\frac{9}{9}$ .

$\frac{(8 + y) (8 - y) \cdot 9}{16 \cdot 9} + \frac{y^{2}}{9} \cdot \frac{16}{16} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.4.2

Multiply $16$ by $9$ .

$\frac{(8 + y) (8 - y) \cdot 9}{144} + \frac{y^{2}}{9} \cdot \frac{16}{16} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.4.3

Multiply $\frac{y^{2}}{9}$ by $\frac{16}{16}$ .

$\frac{(8 + y) (8 - y) \cdot 9}{144} + \frac{y^{2} \cdot 16}{9 \cdot 16} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.4.4

Multiply $9$ by $16$ .

$\frac{(8 + y) (8 - y) \cdot 9}{144} + \frac{y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{(8 + y) (8 - y) \cdot 9}{144} + \frac{y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.5

Combine the numerators over the common denominator.

$\frac{(8 + y) (8 - y) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6

Simplify the numerator.

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Step 2.1.2.1.6.1

Expand $(8 + y) (8 - y)$ using the FOIL Method.

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Step 2.1.2.1.6.1.1

Apply the distributive property.

$\frac{(8 (8 - y) + y (8 - y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.1.2

Apply the distributive property.

$\frac{(8 \cdot 8 + 8 (- y) + y (8 - y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.1.3

Apply the distributive property.

$\frac{(8 \cdot 8 + 8 (- y) + y \cdot 8 + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{(8 \cdot 8 + 8 (- y) + y \cdot 8 + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.2

Simplify and combine like terms.

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Step 2.1.2.1.6.2.1

Simplify each term.

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Step 2.1.2.1.6.2.1.1

Multiply $8$ by $8$ .

$\frac{(64 + 8 (- y) + y \cdot 8 + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.2.1.2

Multiply $- 1$ by $8$ .

$\frac{(64 - 8 y + y \cdot 8 + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.2.1.3

Move $8$ to the left of $y$ .

$\frac{(64 - 8 y + 8 \cdot y + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{(64 - 8 y + 8 y - y \cdot y) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.2.1.6.2.1.5.1

Move $y$ .

$\frac{(64 - 8 y + 8 y - (y \cdot y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.2.1.5.2

Multiply $y$ by $y$ .

$\frac{(64 - 8 y + 8 y - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{(64 - 8 y + 8 y - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{(64 - 8 y + 8 y - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.2.2

Add $- 8 y$ and $8 y$ .

$\frac{(64 + 0 - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.2.3

Add $64$ and $0$ .

$\frac{(64 - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{(64 - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.3

Apply the distributive property.

$\frac{64 \cdot 9 - y^{2} \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.4

Multiply $64$ by $9$ .

$\frac{576 - y^{2} \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.5

Multiply $9$ by $- 1$ .

$\frac{576 - 9 y^{2} + y^{2} \cdot 16}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.6

Move $16$ to the left of $y^{2}$ .

$\frac{576 - 9 y^{2} + 16 \cdot y^{2}}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.1.2.1.6.7

Add $- 9 y^{2}$ and $16 y^{2}$ .

$\frac{576 + 7 y^{2}}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{576 + 7 y^{2}}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{576 + 7 y^{2}}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{576 + 7 y^{2}}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

$\frac{576 + 7 y^{2}}{144} = 1$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2

Solve for $y$ in $\frac{576 + 7 y^{2}}{144} = 1$ .

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Step 2.2.1

Multiply both sides by $144$ .

$\frac{576 + 7 y^{2}}{144} \cdot 144 = 1 \cdot 144$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify the left side.

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Step 2.2.2.1.1

Simplify $\frac{576 + 7 y^{2}}{144} \cdot 144$ .

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Step 2.2.2.1.1.1

Cancel the common factor of $144$ .

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Step 2.2.2.1.1.1.1

Cancel the common factor.

$\frac{576 + 7 y^{2}}{144} \cdot 144 = 1 \cdot 144$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.2.1.1.1.2

Rewrite the expression.

$576 + 7 y^{2} = 1 \cdot 144$

$x = \sqrt{(8 + y) (8 - y)}$

$576 + 7 y^{2} = 1 \cdot 144$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.2.1.1.2

Reorder $576$ and $7 y^{2}$ .

$7 y^{2} + 576 = 1 \cdot 144$

$x = \sqrt{(8 + y) (8 - y)}$

$7 y^{2} + 576 = 1 \cdot 144$

$x = \sqrt{(8 + y) (8 - y)}$

$7 y^{2} + 576 = 1 \cdot 144$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.2.2

Simplify the right side.

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Step 2.2.2.2.1

Multiply $144$ by $1$ .

$7 y^{2} + 576 = 144$

$x = \sqrt{(8 + y) (8 - y)}$

$7 y^{2} + 576 = 144$

$x = \sqrt{(8 + y) (8 - y)}$

$7 y^{2} + 576 = 144$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3

Solve for $y$ .

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Step 2.2.3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.3.1.1

Subtract $576$ from both sides of the equation.

$7 y^{2} = 144 - 576$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.1.2

Subtract $576$ from $144$ .

$7 y^{2} = - 432$

$x = \sqrt{(8 + y) (8 - y)}$

$7 y^{2} = - 432$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.2

Divide each term in $7 y^{2} = - 432$ by $7$ and simplify.

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Step 2.2.3.2.1

Divide each term in $7 y^{2} = - 432$ by $7$ .

$\frac{7 y^{2}}{7} = \frac{- 432}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.2.2

Simplify the left side.

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Step 2.2.3.2.2.1

Cancel the common factor of $7$ .

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Step 2.2.3.2.2.1.1

Cancel the common factor.

$\frac{7 y^{2}}{7} = \frac{- 432}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 432}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

$y^{2} = \frac{- 432}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

$y^{2} = \frac{- 432}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.2.3

Simplify the right side.

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Step 2.2.3.2.3.1

Move the negative in front of the fraction.

$y^{2} = - \frac{432}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

$y^{2} = - \frac{432}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

$y^{2} = - \frac{432}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- \frac{432}{7}}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4

Simplify $\pm \sqrt{- \frac{432}{7}}$ .

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Step 2.2.3.4.1

Rewrite $- 1$ as $i^{2}$ .

$y = \pm \sqrt{i^{2} (\frac{432}{7})}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.2

Pull terms out from under the radical.

$y = \pm i \sqrt{\frac{432}{7}}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.3

Rewrite $\sqrt{\frac{432}{7}}$ as $\frac{\sqrt{432}}{\sqrt{7}}$ .

$y = \pm i (\frac{\sqrt{432}}{\sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.4

Simplify the numerator.

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Step 2.2.3.4.4.1

Rewrite $432$ as $12^{2} \cdot 3$ .

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Step 2.2.3.4.4.1.1

Factor $144$ out of $432$ .

$y = \pm i (\frac{\sqrt{144 (3)}}{\sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.4.1.2

Rewrite $144$ as $12^{2}$ .

$y = \pm i (\frac{\sqrt{12^{2} \cdot 3}}{\sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{\sqrt{12^{2} \cdot 3}}{\sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.4.2

Pull terms out from under the radical.

$y = \pm i (\frac{12 \sqrt{3}}{\sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{3}}{\sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.5

Multiply $\frac{12 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm i (\frac{12 \sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6

Combine and simplify the denominator.

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Step 2.2.3.4.6.1

Multiply $\frac{12 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.2

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.3

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.5

Add $1$ and $1$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.2.3.4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.6.4

Cancel the common factor of $2$ .

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Step 2.2.3.4.6.6.4.1

Cancel the common factor.

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.6.4.2

Rewrite the expression.

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.6.6.5

Evaluate the exponent.

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.7

Simplify the numerator.

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Step 2.2.3.4.7.1

Combine using the product rule for radicals.

$y = \pm i (\frac{12 \sqrt{7 \cdot 3}}{7})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.7.2

Multiply $7$ by $3$ .

$y = \pm i (\frac{12 \sqrt{21}}{7})$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{21}}{7})$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.8

Combine fractions.

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Step 2.2.3.4.8.1

Combine $i$ and $\frac{12 \sqrt{21}}{7}$ .

$y = \pm \frac{i (12 \sqrt{21})}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.4.8.2

Move $12$ to the left of $i$ .

$y = \pm \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \pm \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \pm \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.2.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{12 i \sqrt{21}}{7}, - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \frac{12 i \sqrt{21}}{7}, - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \frac{12 i \sqrt{21}}{7}, - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

$y = \frac{12 i \sqrt{21}}{7}, - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + y) (8 - y)}$

Step 2.3

Replace all occurrences of $y$ with $\frac{12 i \sqrt{21}}{7}$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{(8 + y) (8 - y)}$ with $\frac{12 i \sqrt{21}}{7}$ .

$x = \sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2

Simplify $x = \sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$ .

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Step 2.3.2.1

Simplify the left side.

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Step 2.3.2.1.1

Remove parentheses.

$x = \sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify $\sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$ .

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Step 2.3.2.2.1.1

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \sqrt{(8 \cdot \frac{7}{7} + \frac{12 i \sqrt{21}}{7}) (8 - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.2

Combine $8$ and $\frac{7}{7}$ .

$x = \sqrt{(\frac{8 \cdot 7}{7} + \frac{12 i \sqrt{21}}{7}) (8 - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.3

Combine the numerators over the common denominator.

$x = \sqrt{\frac{56 + 12 i \sqrt{21}}{7} \cdot (8 - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.4

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \sqrt{\frac{56 + 12 i \sqrt{21}}{7} \cdot (8 \cdot \frac{7}{7} - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.5

Combine fractions.

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Step 2.3.2.2.1.5.1

Combine $8$ and $\frac{7}{7}$ .

$x = \sqrt{\frac{56 + 12 i \sqrt{21}}{7} \cdot (\frac{8 \cdot 7}{7} - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.5.2

Combine the numerators over the common denominator.

$x = \sqrt{\frac{56 + 12 i \sqrt{21}}{7} \cdot \frac{56 - 12 i \sqrt{21}}{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.5.3

Multiply $\frac{56 + 12 i \sqrt{21}}{7}$ by $\frac{56 - 12 i \sqrt{21}}{7}$ .

$x = \sqrt{\frac{(56 + 12 i \sqrt{21}) (56 - 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{(56 + 12 i \sqrt{21}) (56 - 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.6

Expand $(56 + 12 i \sqrt{21}) (56 - 12 i \sqrt{21})$ using the FOIL Method.

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Step 2.3.2.2.1.6.1

Apply the distributive property.

$x = \sqrt{\frac{56 (56 - 12 i \sqrt{21}) + 12 i \sqrt{21} (56 - 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.6.2

Apply the distributive property.

$x = \sqrt{\frac{56 \cdot 56 + 56 (- 12 i \sqrt{21}) + 12 i \sqrt{21} (56 - 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.6.3

Apply the distributive property.

$x = \sqrt{\frac{56 \cdot 56 + 56 (- 12 i \sqrt{21}) + 12 i \sqrt{21} \cdot 56 + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{56 \cdot 56 + 56 (- 12 i \sqrt{21}) + 12 i \sqrt{21} \cdot 56 + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7

Simplify and combine like terms.

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Step 2.3.2.2.1.7.1

Simplify each term.

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Step 2.3.2.2.1.7.1.1

Multiply $56$ by $56$ .

$x = \sqrt{\frac{3136 + 56 (- 12 i \sqrt{21}) + 12 i \sqrt{21} \cdot 56 + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.2

Multiply $- 12$ by $56$ .

$x = \sqrt{\frac{3136 - 672 (i \sqrt{21}) + 12 i \sqrt{21} \cdot 56 + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.3

Multiply $56$ by $12$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.4

Multiply $12 i \sqrt{21} (- 12 i \sqrt{21})$ .

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Step 2.3.2.2.1.7.1.4.1

Multiply $- 12$ by $12$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i \sqrt{21} (i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.4.2

Raise $i$ to the power of $1$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 (i i) \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.4.3

Raise $i$ to the power of $1$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 (i i) \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{1 + 1} \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.4.5

Add $1$ and $1$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.4.6

Raise $\sqrt{21}$ to the power of $1$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} (\sqrt{21} \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.4.7

Raise $\sqrt{21}$ to the power of $1$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} (\sqrt{21} \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{1 + 1}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.4.9

Add $1$ and $1$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.5

Rewrite $i^{2}$ as $- 1$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 \cdot (- 1 {\sqrt{21}}^{2})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.6

Multiply $- 144$ by $- 1$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.7

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 2.3.2.2.1.7.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 {(21^{\frac{1}{2}})}^{2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21^{\frac{1}{2} \cdot 2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21^{\frac{2}{2}}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.7.4

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.7.1.7.4.1

Cancel the common factor.

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21^{\frac{2}{2}}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.7.4.2

Rewrite the expression.

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.7.5

Evaluate the exponent.

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.1.8

Multiply $144$ by $21$ .

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 3024}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 3024}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.2

Add $3136$ and $3024$ .

$x = \sqrt{\frac{- 672 i \sqrt{21} + 672 i \sqrt{21} + 6160}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.3

Add $- 672 i \sqrt{21}$ and $672 i \sqrt{21}$ .

$x = \sqrt{\frac{0 + 6160}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.7.4

Add $0$ and $6160$ .

$x = \sqrt{\frac{6160}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{6160}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.8

Cancel the common factor of $6160$ and $7$ .

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Step 2.3.2.2.1.8.1

Factor $7$ out of $6160$ .

$x = \sqrt{\frac{7 \cdot 880}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.8.2

Cancel the common factors.

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Step 2.3.2.2.1.8.2.1

Factor $7$ out of $7 \cdot 7$ .

$x = \sqrt{\frac{7 \cdot 880}{7 (7)}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.8.2.2

Cancel the common factor.

$x = \sqrt{\frac{7 \cdot 880}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.8.2.3

Rewrite the expression.

$x = \sqrt{\frac{880}{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{880}{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{880}{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.9

Rewrite $\sqrt{\frac{880}{7}}$ as $\frac{\sqrt{880}}{\sqrt{7}}$ .

$x = \frac{\sqrt{880}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.10

Simplify the numerator.

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Step 2.3.2.2.1.10.1

Rewrite $880$ as $4^{2} \cdot 55$ .

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Step 2.3.2.2.1.10.1.1

Factor $16$ out of $880$ .

$x = \frac{\sqrt{16 (55)}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.10.1.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{\sqrt{4^{2} \cdot 55}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{\sqrt{4^{2} \cdot 55}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.10.2

Pull terms out from under the radical.

$x = \frac{4 \sqrt{55}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{55}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.11

Multiply $\frac{4 \sqrt{55}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \frac{4 \sqrt{55}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12

Combine and simplify the denominator.

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Step 2.3.2.2.1.12.1

Multiply $\frac{4 \sqrt{55}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{4 \sqrt{55} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.5

Add $1$ and $1$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{{\sqrt{7}}^{2}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.3.2.2.1.12.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.6.4

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.12.6.4.1

Cancel the common factor.

$x = \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.6.4.2

Rewrite the expression.

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.12.6.5

Evaluate the exponent.

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.13

Simplify the numerator.

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Step 2.3.2.2.1.13.1

Combine using the product rule for radicals.

$x = \frac{4 \sqrt{7 \cdot 55}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.3.2.2.1.13.2

Multiply $7$ by $55$ .

$x = \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 2.4

Replace all occurrences of $y$ with $- \frac{12 i \sqrt{21}}{7}$ in each equation.

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Step 2.4.1

Replace all occurrences of $y$ in $x = \sqrt{(8 + y) (8 - y)}$ with $- \frac{12 i \sqrt{21}}{7}$ .

$x = \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2

Simplify $x = \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$ .

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Step 2.4.2.1

Simplify the left side.

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Step 2.4.2.1.1

Remove parentheses.

$x = \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2

Simplify the right side.

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Step 2.4.2.2.1

Simplify $\sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$ .

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Step 2.4.2.2.1.1

Multiply $- - \frac{12 i \sqrt{21}}{7}$ .

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Step 2.4.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$x = \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 + 1 (\frac{12 i \sqrt{21}}{7}))}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.1.2

Multiply $\frac{12 i \sqrt{21}}{7}$ by $1$ .

$x = \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \sqrt{(8 \cdot \frac{7}{7} - \frac{12 i \sqrt{21}}{7}) (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.3

Combine $8$ and $\frac{7}{7}$ .

$x = \sqrt{(\frac{8 \cdot 7}{7} - \frac{12 i \sqrt{21}}{7}) (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.4

Combine the numerators over the common denominator.

$x = \sqrt{\frac{56 - 12 i \sqrt{21}}{7} \cdot (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.5

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \sqrt{\frac{56 - 12 i \sqrt{21}}{7} \cdot (8 \cdot \frac{7}{7} + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.6

Combine fractions.

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Step 2.4.2.2.1.6.1

Combine $8$ and $\frac{7}{7}$ .

$x = \sqrt{\frac{56 - 12 i \sqrt{21}}{7} \cdot (\frac{8 \cdot 7}{7} + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.6.2

Combine the numerators over the common denominator.

$x = \sqrt{\frac{56 - 12 i \sqrt{21}}{7} \cdot \frac{56 + 12 i \sqrt{21}}{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.6.3

Multiply $\frac{56 - 12 i \sqrt{21}}{7}$ by $\frac{56 + 12 i \sqrt{21}}{7}$ .

$x = \sqrt{\frac{(56 - 12 i \sqrt{21}) (56 + 12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{(56 - 12 i \sqrt{21}) (56 + 12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.7

Expand $(56 - 12 i \sqrt{21}) (56 + 12 i \sqrt{21})$ using the FOIL Method.

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Step 2.4.2.2.1.7.1

Apply the distributive property.

$x = \sqrt{\frac{56 (56 + 12 i \sqrt{21}) - 12 i \sqrt{21} (56 + 12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.7.2

Apply the distributive property.

$x = \sqrt{\frac{56 \cdot 56 + 56 (12 i \sqrt{21}) - 12 i \sqrt{21} (56 + 12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.7.3

Apply the distributive property.

$x = \sqrt{\frac{56 \cdot 56 + 56 (12 i \sqrt{21}) - 12 i \sqrt{21} \cdot 56 - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{56 \cdot 56 + 56 (12 i \sqrt{21}) - 12 i \sqrt{21} \cdot 56 - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8

Simplify and combine like terms.

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Step 2.4.2.2.1.8.1

Simplify each term.

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Step 2.4.2.2.1.8.1.1

Multiply $56$ by $56$ .

$x = \sqrt{\frac{3136 + 56 (12 i \sqrt{21}) - 12 i \sqrt{21} \cdot 56 - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.2

Multiply $12$ by $56$ .

$x = \sqrt{\frac{3136 + 672 (i \sqrt{21}) - 12 i \sqrt{21} \cdot 56 - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.3

Multiply $56$ by $- 12$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.4

Multiply $- 12 i \sqrt{21} (12 i \sqrt{21})$ .

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Step 2.4.2.2.1.8.1.4.1

Multiply $12$ by $- 12$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i \sqrt{21} (i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.4.2

Raise $i$ to the power of $1$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 (i i) \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.4.3

Raise $i$ to the power of $1$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 (i i) \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{1 + 1} \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.4.5

Add $1$ and $1$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.4.6

Raise $\sqrt{21}$ to the power of $1$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} (\sqrt{21} \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.4.7

Raise $\sqrt{21}$ to the power of $1$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} (\sqrt{21} \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{1 + 1}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.4.9

Add $1$ and $1$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.5

Rewrite $i^{2}$ as $- 1$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 \cdot (- 1 {\sqrt{21}}^{2})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.6

Multiply $- 144$ by $- 1$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.7

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 2.4.2.2.1.8.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 {(21^{\frac{1}{2}})}^{2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21^{\frac{1}{2} \cdot 2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21^{\frac{2}{2}}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.7.4

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.8.1.7.4.1

Cancel the common factor.

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21^{\frac{2}{2}}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.7.4.2

Rewrite the expression.

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.7.5

Evaluate the exponent.

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.1.8

Multiply $144$ by $21$ .

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 3024}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 3024}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.2

Add $3136$ and $3024$ .

$x = \sqrt{\frac{672 i \sqrt{21} - 672 i \sqrt{21} + 6160}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.3

Subtract $672 i \sqrt{21}$ from $672 i \sqrt{21}$ .

$x = \sqrt{\frac{0 + 6160}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.8.4

Add $0$ and $6160$ .

$x = \sqrt{\frac{6160}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{6160}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.9

Cancel the common factor of $6160$ and $7$ .

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Step 2.4.2.2.1.9.1

Factor $7$ out of $6160$ .

$x = \sqrt{\frac{7 \cdot 880}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.9.2

Cancel the common factors.

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Step 2.4.2.2.1.9.2.1

Factor $7$ out of $7 \cdot 7$ .

$x = \sqrt{\frac{7 \cdot 880}{7 (7)}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.9.2.2

Cancel the common factor.

$x = \sqrt{\frac{7 \cdot 880}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.9.2.3

Rewrite the expression.

$x = \sqrt{\frac{880}{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{880}{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \sqrt{\frac{880}{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.10

Rewrite $\sqrt{\frac{880}{7}}$ as $\frac{\sqrt{880}}{\sqrt{7}}$ .

$x = \frac{\sqrt{880}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.11

Simplify the numerator.

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Step 2.4.2.2.1.11.1

Rewrite $880$ as $4^{2} \cdot 55$ .

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Step 2.4.2.2.1.11.1.1

Factor $16$ out of $880$ .

$x = \frac{\sqrt{16 (55)}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.11.1.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{\sqrt{4^{2} \cdot 55}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{\sqrt{4^{2} \cdot 55}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.11.2

Pull terms out from under the radical.

$x = \frac{4 \sqrt{55}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{55}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.12

Multiply $\frac{4 \sqrt{55}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \frac{4 \sqrt{55}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13

Combine and simplify the denominator.

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Step 2.4.2.2.1.13.1

Multiply $\frac{4 \sqrt{55}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{4 \sqrt{55} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.5

Add $1$ and $1$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{{\sqrt{7}}^{2}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.4.2.2.1.13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.6.4

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.13.6.4.1

Cancel the common factor.

$x = \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.6.4.2

Rewrite the expression.

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.13.6.5

Evaluate the exponent.

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.14

Simplify the numerator.

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Step 2.4.2.2.1.14.1

Combine using the product rule for radicals.

$x = \frac{4 \sqrt{7 \cdot 55}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 2.4.2.2.1.14.2

Multiply $7$ by $55$ .

$x = \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3

Solve the system $x = - \sqrt{(8 + y) (8 - y)}, \frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{(8 + y) (8 - y)}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$ with $- \sqrt{(8 + y) (8 - y)}$ .

$\frac{{(- \sqrt{(8 + y) (8 - y)})}^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $\frac{{(- \sqrt{(8 + y) (8 - y)})}^{2}}{16} + \frac{y^{2}}{9}$ .

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Step 3.1.2.1.1

Simplify the numerator.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{(8 + y) (8 - y)}$ .

$\frac{{(- 1)}^{2} {\sqrt{(8 + y) (8 - y)}}^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.2

Raise $- 1$ to the power of $2$ .

$\frac{1 {\sqrt{(8 + y) (8 - y)}}^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.3

Rewrite ${\sqrt{(8 + y) (8 - y)}}^{2}$ as $(8 + y) (8 - y)$ .

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Step 3.1.2.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(8 + y) (8 - y)}$ as ${((8 + y) (8 - y))}^{\frac{1}{2}}$ .

$\frac{1 {({((8 + y) (8 - y))}^{\frac{1}{2}})}^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1 {((8 + y) (8 - y))}^{\frac{1}{2} \cdot 2}}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{1 {((8 + y) (8 - y))}^{\frac{2}{2}}}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.3.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.3.4.1

Cancel the common factor.

$\frac{1 {((8 + y) (8 - y))}^{\frac{2}{2}}}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.3.4.2

Rewrite the expression.

$\frac{1 ((8 + y) (8 - y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{1 ((8 + y) (8 - y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.3.5

Simplify.

$\frac{1 ((8 + y) (8 - y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{1 ((8 + y) (8 - y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.4

Expand $(8 + y) (8 - y)$ using the FOIL Method.

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Step 3.1.2.1.1.4.1

Apply the distributive property.

$\frac{1 (8 (8 - y) + y (8 - y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.4.2

Apply the distributive property.

$\frac{1 (8 \cdot 8 + 8 (- y) + y (8 - y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.4.3

Apply the distributive property.

$\frac{1 (8 \cdot 8 + 8 (- y) + y \cdot 8 + y (- y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{1 (8 \cdot 8 + 8 (- y) + y \cdot 8 + y (- y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.5

Simplify and combine like terms.

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Step 3.1.2.1.1.5.1

Simplify each term.

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Step 3.1.2.1.1.5.1.1

Multiply $8$ by $8$ .

$\frac{1 (64 + 8 (- y) + y \cdot 8 + y (- y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.5.1.2

Multiply $- 1$ by $8$ .

$\frac{1 (64 - 8 y + y \cdot 8 + y (- y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.5.1.3

Move $8$ to the left of $y$ .

$\frac{1 (64 - 8 y + 8 \cdot y + y (- y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.5.1.4

Rewrite using the commutative property of multiplication.

$\frac{1 (64 - 8 y + 8 y - y \cdot y)}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.5.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.2.1.1.5.1.5.1

Move $y$ .

$\frac{1 (64 - 8 y + 8 y - (y \cdot y))}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.5.1.5.2

Multiply $y$ by $y$ .

$\frac{1 (64 - 8 y + 8 y - y^{2})}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{1 (64 - 8 y + 8 y - y^{2})}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{1 (64 - 8 y + 8 y - y^{2})}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.5.2

Add $- 8 y$ and $8 y$ .

$\frac{1 (64 + 0 - y^{2})}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.5.3

Add $64$ and $0$ .

$\frac{1 (64 - y^{2})}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{1 (64 - y^{2})}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.6

Multiply $64 - y^{2}$ by $1$ .

$\frac{64 - y^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.7

Rewrite $64$ as $8^{2}$ .

$\frac{8^{2} - y^{2}}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.1.8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = y$ .

$\frac{(8 + y) (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{(8 + y) (8 - y)}{16} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.2

To write $\frac{(8 + y) (8 - y)}{16}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{(8 + y) (8 - y)}{16} \cdot \frac{9}{9} + \frac{y^{2}}{9} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.3

To write $\frac{y^{2}}{9}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{(8 + y) (8 - y)}{16} \cdot \frac{9}{9} + \frac{y^{2}}{9} \cdot \frac{16}{16} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.4

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.2.1.4.1

Multiply $\frac{(8 + y) (8 - y)}{16}$ by $\frac{9}{9}$ .

$\frac{(8 + y) (8 - y) \cdot 9}{16 \cdot 9} + \frac{y^{2}}{9} \cdot \frac{16}{16} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.4.2

Multiply $16$ by $9$ .

$\frac{(8 + y) (8 - y) \cdot 9}{144} + \frac{y^{2}}{9} \cdot \frac{16}{16} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.4.3

Multiply $\frac{y^{2}}{9}$ by $\frac{16}{16}$ .

$\frac{(8 + y) (8 - y) \cdot 9}{144} + \frac{y^{2} \cdot 16}{9 \cdot 16} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.4.4

Multiply $9$ by $16$ .

$\frac{(8 + y) (8 - y) \cdot 9}{144} + \frac{y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{(8 + y) (8 - y) \cdot 9}{144} + \frac{y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.5

Combine the numerators over the common denominator.

$\frac{(8 + y) (8 - y) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6

Simplify the numerator.

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Step 3.1.2.1.6.1

Expand $(8 + y) (8 - y)$ using the FOIL Method.

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Step 3.1.2.1.6.1.1

Apply the distributive property.

$\frac{(8 (8 - y) + y (8 - y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.1.2

Apply the distributive property.

$\frac{(8 \cdot 8 + 8 (- y) + y (8 - y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.1.3

Apply the distributive property.

$\frac{(8 \cdot 8 + 8 (- y) + y \cdot 8 + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{(8 \cdot 8 + 8 (- y) + y \cdot 8 + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.2

Simplify and combine like terms.

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Step 3.1.2.1.6.2.1

Simplify each term.

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Step 3.1.2.1.6.2.1.1

Multiply $8$ by $8$ .

$\frac{(64 + 8 (- y) + y \cdot 8 + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.2.1.2

Multiply $- 1$ by $8$ .

$\frac{(64 - 8 y + y \cdot 8 + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.2.1.3

Move $8$ to the left of $y$ .

$\frac{(64 - 8 y + 8 \cdot y + y (- y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{(64 - 8 y + 8 y - y \cdot y) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.2.1.6.2.1.5.1

Move $y$ .

$\frac{(64 - 8 y + 8 y - (y \cdot y)) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.2.1.5.2

Multiply $y$ by $y$ .

$\frac{(64 - 8 y + 8 y - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{(64 - 8 y + 8 y - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{(64 - 8 y + 8 y - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.2.2

Add $- 8 y$ and $8 y$ .

$\frac{(64 + 0 - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.2.3

Add $64$ and $0$ .

$\frac{(64 - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{(64 - y^{2}) \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.3

Apply the distributive property.

$\frac{64 \cdot 9 - y^{2} \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.4

Multiply $64$ by $9$ .

$\frac{576 - y^{2} \cdot 9 + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.5

Multiply $9$ by $- 1$ .

$\frac{576 - 9 y^{2} + y^{2} \cdot 16}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.6

Move $16$ to the left of $y^{2}$ .

$\frac{576 - 9 y^{2} + 16 \cdot y^{2}}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.1.2.1.6.7

Add $- 9 y^{2}$ and $16 y^{2}$ .

$\frac{576 + 7 y^{2}}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{576 + 7 y^{2}}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{576 + 7 y^{2}}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{576 + 7 y^{2}}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

$\frac{576 + 7 y^{2}}{144} = 1$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2

Solve for $y$ in $\frac{576 + 7 y^{2}}{144} = 1$ .

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Step 3.2.1

Multiply both sides by $144$ .

$\frac{576 + 7 y^{2}}{144} \cdot 144 = 1 \cdot 144$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Simplify $\frac{576 + 7 y^{2}}{144} \cdot 144$ .

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Step 3.2.2.1.1.1

Cancel the common factor of $144$ .

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Step 3.2.2.1.1.1.1

Cancel the common factor.

$\frac{576 + 7 y^{2}}{144} \cdot 144 = 1 \cdot 144$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.2.1.1.1.2

Rewrite the expression.

$576 + 7 y^{2} = 1 \cdot 144$

$x = - \sqrt{(8 + y) (8 - y)}$

$576 + 7 y^{2} = 1 \cdot 144$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.2.1.1.2

Reorder $576$ and $7 y^{2}$ .

$7 y^{2} + 576 = 1 \cdot 144$

$x = - \sqrt{(8 + y) (8 - y)}$

$7 y^{2} + 576 = 1 \cdot 144$

$x = - \sqrt{(8 + y) (8 - y)}$

$7 y^{2} + 576 = 1 \cdot 144$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Multiply $144$ by $1$ .

$7 y^{2} + 576 = 144$

$x = - \sqrt{(8 + y) (8 - y)}$

$7 y^{2} + 576 = 144$

$x = - \sqrt{(8 + y) (8 - y)}$

$7 y^{2} + 576 = 144$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3

Solve for $y$ .

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Step 3.2.3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.2.3.1.1

Subtract $576$ from both sides of the equation.

$7 y^{2} = 144 - 576$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.1.2

Subtract $576$ from $144$ .

$7 y^{2} = - 432$

$x = - \sqrt{(8 + y) (8 - y)}$

$7 y^{2} = - 432$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.2

Divide each term in $7 y^{2} = - 432$ by $7$ and simplify.

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Step 3.2.3.2.1

Divide each term in $7 y^{2} = - 432$ by $7$ .

$\frac{7 y^{2}}{7} = \frac{- 432}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.2.2

Simplify the left side.

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Step 3.2.3.2.2.1

Cancel the common factor of $7$ .

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Step 3.2.3.2.2.1.1

Cancel the common factor.

$\frac{7 y^{2}}{7} = \frac{- 432}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 432}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

$y^{2} = \frac{- 432}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

$y^{2} = \frac{- 432}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.2.3

Simplify the right side.

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Step 3.2.3.2.3.1

Move the negative in front of the fraction.

$y^{2} = - \frac{432}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

$y^{2} = - \frac{432}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

$y^{2} = - \frac{432}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- \frac{432}{7}}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4

Simplify $\pm \sqrt{- \frac{432}{7}}$ .

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Step 3.2.3.4.1

Rewrite $- 1$ as $i^{2}$ .

$y = \pm \sqrt{i^{2} (\frac{432}{7})}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.2

Pull terms out from under the radical.

$y = \pm i \sqrt{\frac{432}{7}}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.3

Rewrite $\sqrt{\frac{432}{7}}$ as $\frac{\sqrt{432}}{\sqrt{7}}$ .

$y = \pm i (\frac{\sqrt{432}}{\sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.4

Simplify the numerator.

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Step 3.2.3.4.4.1

Rewrite $432$ as $12^{2} \cdot 3$ .

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Step 3.2.3.4.4.1.1

Factor $144$ out of $432$ .

$y = \pm i (\frac{\sqrt{144 (3)}}{\sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.4.1.2

Rewrite $144$ as $12^{2}$ .

$y = \pm i (\frac{\sqrt{12^{2} \cdot 3}}{\sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{\sqrt{12^{2} \cdot 3}}{\sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.4.2

Pull terms out from under the radical.

$y = \pm i (\frac{12 \sqrt{3}}{\sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{3}}{\sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.5

Multiply $\frac{12 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm i (\frac{12 \sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6

Combine and simplify the denominator.

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Step 3.2.3.4.6.1

Multiply $\frac{12 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.2

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.3

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.5

Add $1$ and $1$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.2.3.4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.6.4

Cancel the common factor of $2$ .

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Step 3.2.3.4.6.6.4.1

Cancel the common factor.

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.6.4.2

Rewrite the expression.

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.6.6.5

Evaluate the exponent.

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{3} \sqrt{7}}{7})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.7

Simplify the numerator.

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Step 3.2.3.4.7.1

Combine using the product rule for radicals.

$y = \pm i (\frac{12 \sqrt{7 \cdot 3}}{7})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.7.2

Multiply $7$ by $3$ .

$y = \pm i (\frac{12 \sqrt{21}}{7})$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \pm i (\frac{12 \sqrt{21}}{7})$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.8

Combine fractions.

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Step 3.2.3.4.8.1

Combine $i$ and $\frac{12 \sqrt{21}}{7}$ .

$y = \pm \frac{i (12 \sqrt{21})}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.4.8.2

Move $12$ to the left of $i$ .

$y = \pm \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \pm \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \pm \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.2.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{12 i \sqrt{21}}{7}, - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \frac{12 i \sqrt{21}}{7}, - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \frac{12 i \sqrt{21}}{7}, - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

$y = \frac{12 i \sqrt{21}}{7}, - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + y) (8 - y)}$

Step 3.3

Replace all occurrences of $y$ with $\frac{12 i \sqrt{21}}{7}$ in each equation.

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \sqrt{(8 + y) (8 - y)}$ with $\frac{12 i \sqrt{21}}{7}$ .

$x = - \sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2

Simplify $x = - \sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$ .

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Remove parentheses.

$x = - \sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Simplify $- \sqrt{(8 + \frac{12 i \sqrt{21}}{7}) (8 - (\frac{12 i \sqrt{21}}{7}))}$ .

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Step 3.3.2.2.1.1

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = - \sqrt{(8 \cdot \frac{7}{7} + \frac{12 i \sqrt{21}}{7}) (8 - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.2

Combine $8$ and $\frac{7}{7}$ .

$x = - \sqrt{(\frac{8 \cdot 7}{7} + \frac{12 i \sqrt{21}}{7}) (8 - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.3

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{56 + 12 i \sqrt{21}}{7} \cdot (8 - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.4

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = - \sqrt{\frac{56 + 12 i \sqrt{21}}{7} \cdot (8 \cdot \frac{7}{7} - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.5

Combine fractions.

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Step 3.3.2.2.1.5.1

Combine $8$ and $\frac{7}{7}$ .

$x = - \sqrt{\frac{56 + 12 i \sqrt{21}}{7} \cdot (\frac{8 \cdot 7}{7} - \frac{12 i \sqrt{21}}{7})}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.5.2

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{56 + 12 i \sqrt{21}}{7} \cdot \frac{56 - 12 i \sqrt{21}}{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.5.3

Multiply $\frac{56 + 12 i \sqrt{21}}{7}$ by $\frac{56 - 12 i \sqrt{21}}{7}$ .

$x = - \sqrt{\frac{(56 + 12 i \sqrt{21}) (56 - 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{(56 + 12 i \sqrt{21}) (56 - 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.6

Expand $(56 + 12 i \sqrt{21}) (56 - 12 i \sqrt{21})$ using the FOIL Method.

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Step 3.3.2.2.1.6.1

Apply the distributive property.

$x = - \sqrt{\frac{56 (56 - 12 i \sqrt{21}) + 12 i \sqrt{21} (56 - 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.6.2

Apply the distributive property.

$x = - \sqrt{\frac{56 \cdot 56 + 56 (- 12 i \sqrt{21}) + 12 i \sqrt{21} (56 - 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.6.3

Apply the distributive property.

$x = - \sqrt{\frac{56 \cdot 56 + 56 (- 12 i \sqrt{21}) + 12 i \sqrt{21} \cdot 56 + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{56 \cdot 56 + 56 (- 12 i \sqrt{21}) + 12 i \sqrt{21} \cdot 56 + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7

Simplify and combine like terms.

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Step 3.3.2.2.1.7.1

Simplify each term.

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Step 3.3.2.2.1.7.1.1

Multiply $56$ by $56$ .

$x = - \sqrt{\frac{3136 + 56 (- 12 i \sqrt{21}) + 12 i \sqrt{21} \cdot 56 + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.2

Multiply $- 12$ by $56$ .

$x = - \sqrt{\frac{3136 - 672 (i \sqrt{21}) + 12 i \sqrt{21} \cdot 56 + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.3

Multiply $56$ by $12$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 12 i \sqrt{21} (- 12 i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.4

Multiply $12 i \sqrt{21} (- 12 i \sqrt{21})$ .

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Step 3.3.2.2.1.7.1.4.1

Multiply $- 12$ by $12$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i \sqrt{21} (i \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.4.2

Raise $i$ to the power of $1$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 (i i) \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.4.3

Raise $i$ to the power of $1$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 (i i) \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{1 + 1} \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.4.5

Add $1$ and $1$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.4.6

Raise $\sqrt{21}$ to the power of $1$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} (\sqrt{21} \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.4.7

Raise $\sqrt{21}$ to the power of $1$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} (\sqrt{21} \sqrt{21})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{1 + 1}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.4.9

Add $1$ and $1$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.5

Rewrite $i^{2}$ as $- 1$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} - 144 \cdot (- 1 {\sqrt{21}}^{2})}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.6

Multiply $- 144$ by $- 1$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.7

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 3.3.2.2.1.7.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 {(21^{\frac{1}{2}})}^{2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21^{\frac{1}{2} \cdot 2}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21^{\frac{2}{2}}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.7.4

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.7.1.7.4.1

Cancel the common factor.

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21^{\frac{2}{2}}}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.7.4.2

Rewrite the expression.

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.7.5

Evaluate the exponent.

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.1.8

Multiply $144$ by $21$ .

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 3024}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{3136 - 672 i \sqrt{21} + 672 i \sqrt{21} + 3024}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.2

Add $3136$ and $3024$ .

$x = - \sqrt{\frac{- 672 i \sqrt{21} + 672 i \sqrt{21} + 6160}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.3

Add $- 672 i \sqrt{21}$ and $672 i \sqrt{21}$ .

$x = - \sqrt{\frac{0 + 6160}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.7.4

Add $0$ and $6160$ .

$x = - \sqrt{\frac{6160}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{6160}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.8

Cancel the common factor of $6160$ and $7$ .

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Step 3.3.2.2.1.8.1

Factor $7$ out of $6160$ .

$x = - \sqrt{\frac{7 \cdot 880}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.8.2

Cancel the common factors.

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Step 3.3.2.2.1.8.2.1

Factor $7$ out of $7 \cdot 7$ .

$x = - \sqrt{\frac{7 \cdot 880}{7 (7)}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.8.2.2

Cancel the common factor.

$x = - \sqrt{\frac{7 \cdot 880}{7 \cdot 7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.8.2.3

Rewrite the expression.

$x = - \sqrt{\frac{880}{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{880}{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{880}{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.9

Rewrite $\sqrt{\frac{880}{7}}$ as $\frac{\sqrt{880}}{\sqrt{7}}$ .

$x = - \frac{\sqrt{880}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.10

Simplify the numerator.

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Step 3.3.2.2.1.10.1

Rewrite $880$ as $4^{2} \cdot 55$ .

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Step 3.3.2.2.1.10.1.1

Factor $16$ out of $880$ .

$x = - \frac{\sqrt{16 (55)}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.10.1.2

Rewrite $16$ as $4^{2}$ .

$x = - \frac{\sqrt{4^{2} \cdot 55}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{\sqrt{4^{2} \cdot 55}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.10.2

Pull terms out from under the radical.

$x = - \frac{4 \sqrt{55}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{55}}{\sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.11

Multiply $\frac{4 \sqrt{55}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = - (\frac{4 \sqrt{55}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12

Combine and simplify the denominator.

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Step 3.3.2.2.1.12.1

Multiply $\frac{4 \sqrt{55}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = - \frac{4 \sqrt{55} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.5

Add $1$ and $1$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{{\sqrt{7}}^{2}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.3.2.2.1.12.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.6.4

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.12.6.4.1

Cancel the common factor.

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.6.4.2

Rewrite the expression.

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.12.6.5

Evaluate the exponent.

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.13

Simplify the numerator.

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Step 3.3.2.2.1.13.1

Combine using the product rule for radicals.

$x = - \frac{4 \sqrt{7 \cdot 55}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.3.2.2.1.13.2

Multiply $7$ by $55$ .

$x = - \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = \frac{12 i \sqrt{21}}{7}$

Step 3.4

Replace all occurrences of $y$ with $- \frac{12 i \sqrt{21}}{7}$ in each equation.

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Step 3.4.1

Replace all occurrences of $y$ in $x = - \sqrt{(8 + y) (8 - y)}$ with $- \frac{12 i \sqrt{21}}{7}$ .

$x = - \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2

Simplify $x = - \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$ .

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Step 3.4.2.1

Simplify the left side.

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Step 3.4.2.1.1

Remove parentheses.

$x = - \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2

Simplify the right side.

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Step 3.4.2.2.1

Simplify $- \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 - (- \frac{12 i \sqrt{21}}{7}))}$ .

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Step 3.4.2.2.1.1

Multiply $- - \frac{12 i \sqrt{21}}{7}$ .

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Step 3.4.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$x = - \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 + 1 (\frac{12 i \sqrt{21}}{7}))}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.1.2

Multiply $\frac{12 i \sqrt{21}}{7}$ by $1$ .

$x = - \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{(8 - \frac{12 i \sqrt{21}}{7}) (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = - \sqrt{(8 \cdot \frac{7}{7} - \frac{12 i \sqrt{21}}{7}) (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.3

Combine $8$ and $\frac{7}{7}$ .

$x = - \sqrt{(\frac{8 \cdot 7}{7} - \frac{12 i \sqrt{21}}{7}) (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.4

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{56 - 12 i \sqrt{21}}{7} \cdot (8 + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.5

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = - \sqrt{\frac{56 - 12 i \sqrt{21}}{7} \cdot (8 \cdot \frac{7}{7} + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.6

Combine fractions.

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Step 3.4.2.2.1.6.1

Combine $8$ and $\frac{7}{7}$ .

$x = - \sqrt{\frac{56 - 12 i \sqrt{21}}{7} \cdot (\frac{8 \cdot 7}{7} + \frac{12 i \sqrt{21}}{7})}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.6.2

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{56 - 12 i \sqrt{21}}{7} \cdot \frac{56 + 12 i \sqrt{21}}{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.6.3

Multiply $\frac{56 - 12 i \sqrt{21}}{7}$ by $\frac{56 + 12 i \sqrt{21}}{7}$ .

$x = - \sqrt{\frac{(56 - 12 i \sqrt{21}) (56 + 12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{(56 - 12 i \sqrt{21}) (56 + 12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.7

Expand $(56 - 12 i \sqrt{21}) (56 + 12 i \sqrt{21})$ using the FOIL Method.

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Step 3.4.2.2.1.7.1

Apply the distributive property.

$x = - \sqrt{\frac{56 (56 + 12 i \sqrt{21}) - 12 i \sqrt{21} (56 + 12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.7.2

Apply the distributive property.

$x = - \sqrt{\frac{56 \cdot 56 + 56 (12 i \sqrt{21}) - 12 i \sqrt{21} (56 + 12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.7.3

Apply the distributive property.

$x = - \sqrt{\frac{56 \cdot 56 + 56 (12 i \sqrt{21}) - 12 i \sqrt{21} \cdot 56 - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{56 \cdot 56 + 56 (12 i \sqrt{21}) - 12 i \sqrt{21} \cdot 56 - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8

Simplify and combine like terms.

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Step 3.4.2.2.1.8.1

Simplify each term.

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Step 3.4.2.2.1.8.1.1

Multiply $56$ by $56$ .

$x = - \sqrt{\frac{3136 + 56 (12 i \sqrt{21}) - 12 i \sqrt{21} \cdot 56 - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.2

Multiply $12$ by $56$ .

$x = - \sqrt{\frac{3136 + 672 (i \sqrt{21}) - 12 i \sqrt{21} \cdot 56 - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.3

Multiply $56$ by $- 12$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 12 i \sqrt{21} (12 i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.4

Multiply $- 12 i \sqrt{21} (12 i \sqrt{21})$ .

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Step 3.4.2.2.1.8.1.4.1

Multiply $12$ by $- 12$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i \sqrt{21} (i \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.4.2

Raise $i$ to the power of $1$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 (i i) \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.4.3

Raise $i$ to the power of $1$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 (i i) \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{1 + 1} \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.4.5

Add $1$ and $1$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} \sqrt{21} \sqrt{21}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.4.6

Raise $\sqrt{21}$ to the power of $1$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} (\sqrt{21} \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.4.7

Raise $\sqrt{21}$ to the power of $1$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} (\sqrt{21} \sqrt{21})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{1 + 1}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.4.9

Add $1$ and $1$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 i^{2} {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.5

Rewrite $i^{2}$ as $- 1$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} - 144 \cdot (- 1 {\sqrt{21}}^{2})}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.6

Multiply $- 144$ by $- 1$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 {\sqrt{21}}^{2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.7

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 3.4.2.2.1.8.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 {(21^{\frac{1}{2}})}^{2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21^{\frac{1}{2} \cdot 2}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21^{\frac{2}{2}}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.7.4

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.8.1.7.4.1

Cancel the common factor.

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21^{\frac{2}{2}}}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.7.4.2

Rewrite the expression.

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.7.5

Evaluate the exponent.

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 144 \cdot 21}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.1.8

Multiply $144$ by $21$ .

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 3024}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{3136 + 672 i \sqrt{21} - 672 i \sqrt{21} + 3024}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.2

Add $3136$ and $3024$ .

$x = - \sqrt{\frac{672 i \sqrt{21} - 672 i \sqrt{21} + 6160}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.3

Subtract $672 i \sqrt{21}$ from $672 i \sqrt{21}$ .

$x = - \sqrt{\frac{0 + 6160}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.8.4

Add $0$ and $6160$ .

$x = - \sqrt{\frac{6160}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{6160}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.9

Cancel the common factor of $6160$ and $7$ .

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Step 3.4.2.2.1.9.1

Factor $7$ out of $6160$ .

$x = - \sqrt{\frac{7 \cdot 880}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.9.2

Cancel the common factors.

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Step 3.4.2.2.1.9.2.1

Factor $7$ out of $7 \cdot 7$ .

$x = - \sqrt{\frac{7 \cdot 880}{7 (7)}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.9.2.2

Cancel the common factor.

$x = - \sqrt{\frac{7 \cdot 880}{7 \cdot 7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.9.2.3

Rewrite the expression.

$x = - \sqrt{\frac{880}{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{880}{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \sqrt{\frac{880}{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.10

Rewrite $\sqrt{\frac{880}{7}}$ as $\frac{\sqrt{880}}{\sqrt{7}}$ .

$x = - \frac{\sqrt{880}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.11

Simplify the numerator.

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Step 3.4.2.2.1.11.1

Rewrite $880$ as $4^{2} \cdot 55$ .

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Step 3.4.2.2.1.11.1.1

Factor $16$ out of $880$ .

$x = - \frac{\sqrt{16 (55)}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.11.1.2

Rewrite $16$ as $4^{2}$ .

$x = - \frac{\sqrt{4^{2} \cdot 55}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{\sqrt{4^{2} \cdot 55}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.11.2

Pull terms out from under the radical.

$x = - \frac{4 \sqrt{55}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{55}}{\sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.12

Multiply $\frac{4 \sqrt{55}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = - (\frac{4 \sqrt{55}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13

Combine and simplify the denominator.

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Step 3.4.2.2.1.13.1

Multiply $\frac{4 \sqrt{55}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = - \frac{4 \sqrt{55} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.5

Add $1$ and $1$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{{\sqrt{7}}^{2}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.4.2.2.1.13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.6.4

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.13.6.4.1

Cancel the common factor.

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7^{\frac{2}{2}}}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.6.4.2

Rewrite the expression.

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.13.6.5

Evaluate the exponent.

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{55} \sqrt{7}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.14

Simplify the numerator.

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Step 3.4.2.2.1.14.1

Combine using the product rule for radicals.

$x = - \frac{4 \sqrt{7 \cdot 55}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 3.4.2.2.1.14.2

Multiply $7$ by $55$ .

$x = - \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}$

$y = - \frac{12 i \sqrt{21}}{7}$

Step 4

List all of the solutions.

$x = \frac{4 \sqrt{385}}{7}, y = \frac{12 i \sqrt{21}}{7}$

$x = \frac{4 \sqrt{385}}{7}, y = - \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}, y = \frac{12 i \sqrt{21}}{7}$

$x = - \frac{4 \sqrt{385}}{7}, y = - \frac{12 i \sqrt{21}}{7}$

Step 5