Basic Math Examples

$y = \frac{4}{\sqrt{16 - z^{2}}}$

Step 1

Rewrite the equation as $\frac{4}{\sqrt{16 - z^{2}}} = y$ .

$\frac{4}{\sqrt{16 - z^{2}}} = y$

Step 2

Cross multiply.

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Step 2.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$y \cdot (\sqrt{16 - z^{2}}) = 4$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $y \cdot (\sqrt{16 - z^{2}})$ .

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Step 2.2.1.1

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

$y \cdot \sqrt{4^{2} - z^{2}} = 4$

Step 2.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = z$ .

$y \sqrt{(4 + z) (4 - z)} = 4$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${(y \sqrt{(4 + z) (4 - z)})}^{2} = 4^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 + z) (4 - z)}$ as ${((4 + z) (4 - z))}^{\frac{1}{2}}$ .

${(y {((4 + z) (4 - z))}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(y {((4 + z) (4 - z))}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Expand $(4 + z) (4 - z)$ using the FOIL Method.

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Step 4.2.1.1.1

Apply the distributive property.

${(y {(4 (4 - z) + z (4 - z))}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.1.2

Apply the distributive property.

${(y {(4 \cdot 4 + 4 (- z) + z (4 - z))}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.1.3

Apply the distributive property.

${(y {(4 \cdot 4 + 4 (- z) + z \cdot 4 + z (- z))}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.2

Simplify and combine like terms.

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Step 4.2.1.2.1

Simplify each term.

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Step 4.2.1.2.1.1

Multiply $4$ by $4$ .

${(y {(16 + 4 (- z) + z \cdot 4 + z (- z))}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.2.1.2

Multiply $- 1$ by $4$ .

${(y {(16 - 4 z + z \cdot 4 + z (- z))}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.2.1.3

Move $4$ to the left of $z$ .

${(y {(16 - 4 z + 4 \cdot z + z (- z))}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

${(y {(16 - 4 z + 4 z - z \cdot z)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.2.1.5

Multiply $z$ by $z$ by adding the exponents.

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Step 4.2.1.2.1.5.1

Move $z$ .

${(y {(16 - 4 z + 4 z - (z \cdot z))}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.2.1.5.2

Multiply $z$ by $z$ .

${(y {(16 - 4 z + 4 z - z^{2})}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.2.2

Add $- 4 z$ and $4 z$ .

${(y {(16 + 0 - z^{2})}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.2.3

Add $16$ and $0$ .

${(y {(16 - z^{2})}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.3

Apply the product rule to $y {(16 - z^{2})}^{\frac{1}{2}}$ .

$y^{2} {({(16 - z^{2})}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.4

Multiply the exponents in ${({(16 - z^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.4.1

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

$y^{2} {(16 - z^{2})}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

$y^{2} {(16 - z^{2})}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 4.2.1.4.2.2

Rewrite the expression.

$y^{2} {(16 - z^{2})}^{1} = 4^{2}$

Step 4.2.1.5

Simplify.

$y^{2} (16 - z^{2}) = 4^{2}$

Step 4.2.1.6

Apply the distributive property.

$y^{2} \cdot 16 + y^{2} (- z^{2}) = 4^{2}$

Step 4.2.1.7

Reorder.

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Step 4.2.1.7.1

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

$16 \cdot y^{2} + y^{2} (- z^{2}) = 4^{2}$

Step 4.2.1.7.2

Rewrite using the commutative property of multiplication.

$16 y^{2} - y^{2} z^{2} = 4^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Raise $4$ to the power of $2$ .

$16 y^{2} - y^{2} z^{2} = 16$

Step 5

Solve for $z$ .

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Step 5.1

Subtract $16 y^{2}$ from both sides of the equation.

$- y^{2} z^{2} = 16 - 16 y^{2}$

Step 5.2

Divide each term in $- y^{2} z^{2} = 16 - 16 y^{2}$ by $- y^{2}$ and simplify.

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Step 5.2.1

Divide each term in $- y^{2} z^{2} = 16 - 16 y^{2}$ by $- y^{2}$ .

$\frac{- y^{2} z^{2}}{- y^{2}} = \frac{16}{- y^{2}} + \frac{- 16 y^{2}}{- y^{2}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2} z^{2}}{y^{2}} = \frac{16}{- y^{2}} + \frac{- 16 y^{2}}{- y^{2}}$

Step 5.2.2.2

Cancel the common factor of $y^{2}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{y^{2} z^{2}}{y^{2}} = \frac{16}{- y^{2}} + \frac{- 16 y^{2}}{- y^{2}}$

Step 5.2.2.2.2

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

$z^{2} = \frac{16}{- y^{2}} + \frac{- 16 y^{2}}{- y^{2}}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative in front of the fraction.

$z^{2} = - \frac{16}{y^{2}} + \frac{- 16 y^{2}}{- y^{2}}$

Step 5.2.3.1.2

Cancel the common factor of $y^{2}$ .

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Step 5.2.3.1.2.1

Cancel the common factor.

$z^{2} = - \frac{16}{y^{2}} + \frac{- 16 y^{2}}{- y^{2}}$

Step 5.2.3.1.2.2

Rewrite the expression.

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

Step 5.2.3.1.2.3

Move the negative one from the denominator of $\frac{- 16}{- 1}$ .

$z^{2} = - \frac{16}{y^{2}} - 1 \cdot - 16$

Step 5.2.3.1.3

Rewrite $- 1 \cdot - 16$ as $- - 16$ .

$z^{2} = - \frac{16}{y^{2}} - - 16$

Step 5.2.3.1.4

Multiply $- 1$ by $- 16$ .

$z^{2} = - \frac{16}{y^{2}} + 16$

Step 5.3

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

$z = \pm \sqrt{- \frac{16}{y^{2}} + 16}$

Step 5.4

Simplify $\pm \sqrt{- \frac{16}{y^{2}} + 16}$ .

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Step 5.4.1

Factor $16$ out of $- \frac{16}{y^{2}} + 16$ .

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Step 5.4.1.1

Factor $16$ out of $- \frac{16}{y^{2}}$ .

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

Step 5.4.1.2

Factor $16$ out of $16$ .

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

Step 5.4.1.3

Factor $16$ out of $16 (- \frac{1}{y^{2}}) + 16 (1)$ .

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

Step 5.4.2

Simplify the expression.

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Step 5.4.2.1

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

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

Step 5.4.2.2

Rewrite $\frac{1}{y^{2}}$ as ${(\frac{1}{y})}^{2}$ .

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

Step 5.4.2.3

Reorder $- {(\frac{1}{y})}^{2}$ and $1^{2}$ .

$z = \pm \sqrt{16 (1^{2} - {(\frac{1}{y})}^{2})}$

Step 5.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{1}{y}$ .

$z = \pm \sqrt{16 (1 + \frac{1}{y}) (1 - \frac{1}{y})}$

Step 5.4.4

Write $1$ as a fraction with a common denominator.

$z = \pm \sqrt{16 (\frac{y}{y} + \frac{1}{y}) (1 - \frac{1}{y})}$

Step 5.4.5

Combine the numerators over the common denominator.

$z = \pm \sqrt{16 \frac{y + 1}{y} (1 - \frac{1}{y})}$

Step 5.4.6

Write $1$ as a fraction with a common denominator.

$z = \pm \sqrt{16 \frac{y + 1}{y} (\frac{y}{y} - \frac{1}{y})}$

Step 5.4.7

Combine the numerators over the common denominator.

$z = \pm \sqrt{16 \frac{y + 1}{y} \frac{y - 1}{y}}$

Step 5.4.8

Combine exponents.

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Step 5.4.8.1

Combine $16$ and $\frac{y + 1}{y}$ .

$z = \pm \sqrt{\frac{16 (y + 1)}{y} \cdot \frac{y - 1}{y}}$

Step 5.4.8.2

Multiply $\frac{16 (y + 1)}{y}$ by $\frac{y - 1}{y}$ .

$z = \pm \sqrt{\frac{16 (y + 1) (y - 1)}{y \cdot y}}$

Step 5.4.8.3

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

$z = \pm \sqrt{\frac{16 (y + 1) (y - 1)}{y^{1} y}}$

Step 5.4.8.4

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

$z = \pm \sqrt{\frac{16 (y + 1) (y - 1)}{y^{1} y^{1}}}$

Step 5.4.8.5

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

$z = \pm \sqrt{\frac{16 (y + 1) (y - 1)}{y^{1 + 1}}}$

Step 5.4.8.6

Add $1$ and $1$ .

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

Step 5.4.9

Rewrite $\frac{16 (y + 1) (y - 1)}{y^{2}}$ as ${(\frac{2^{2}}{y})}^{2} ((y + 1) (y - 1))$ .

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Step 5.4.9.1

Factor the perfect power ${(2^{2})}^{2}$ out of $16 (y + 1) (y - 1)$ .

$z = \pm \sqrt{\frac{{(2^{2})}^{2} ((y + 1) (y - 1))}{y^{2}}}$

Step 5.4.9.2

Factor the perfect power $y^{2}$ out of $y^{2}$ .

$z = \pm \sqrt{\frac{{(2^{2})}^{2} ((y + 1) (y - 1))}{y^{2} \cdot 1}}$

Step 5.4.9.3

Rearrange the fraction $\frac{{(2^{2})}^{2} ((y + 1) (y - 1))}{y^{2} \cdot 1}$ .

$z = \pm \sqrt{{(\frac{2^{2}}{y})}^{2} ((y + 1) (y - 1))}$

Step 5.4.10

Pull terms out from under the radical.

$z = \pm \frac{2^{2}}{y} \sqrt{(y + 1) (y - 1)}$

Step 5.4.11

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

$z = \pm \frac{4}{y} \sqrt{(y + 1) (y - 1)}$

Step 5.4.12

Combine $\frac{4}{y}$ and $\sqrt{(y + 1) (y - 1)}$ .

$z = \pm \frac{4 \sqrt{(y + 1) (y - 1)}}{y}$

Step 5.5

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

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Step 5.5.1

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

$z = \frac{4 \sqrt{(y + 1) (y - 1)}}{y}$

Step 5.5.2

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

$z = - \frac{4 \sqrt{(y + 1) (y - 1)}}{y}$

Step 5.5.3

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

$z = \frac{4 \sqrt{(y + 1) (y - 1)}}{y}$

$z = - \frac{4 \sqrt{(y + 1) (y - 1)}}{y}$

$z = \frac{4 \sqrt{(y + 1) (y - 1)}}{y}$

$z = - \frac{4 \sqrt{(y + 1) (y - 1)}}{y}$

$z = \frac{4 \sqrt{(y + 1) (y - 1)}}{y}$

$z = - \frac{4 \sqrt{(y + 1) (y - 1)}}{y}$