Precalculus Examples

${(- \frac{1}{3})}^{2} + y^{2} = 1$

Step 1

Simplify each term.

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Step 1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.1

Apply the product rule to $- \frac{1}{3}$ .

${(- 1)}^{2} {(\frac{1}{3})}^{2} + y^{2} = 1$

Step 1.1.2

Apply the product rule to $\frac{1}{3}$ .

${(- 1)}^{2} \frac{1^{2}}{3^{2}} + y^{2} = 1$

Step 1.2

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

$1 \frac{1^{2}}{3^{2}} + y^{2} = 1$

Step 1.3

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

$\frac{1^{2}}{3^{2}} + y^{2} = 1$

Step 1.4

One to any power is one.

$\frac{1}{3^{2}} + y^{2} = 1$

Step 1.5

Raise $3$ to the power of $2$ .

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

Step 2

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

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Step 2.1

Subtract $\frac{1}{9}$ from both sides of the equation.

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Subtract $1$ from $9$ .

$y^{2} = \frac{8}{9}$

Step 3

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

$y = \pm \sqrt{\frac{8}{9}}$

Step 4

Simplify $\pm \sqrt{\frac{8}{9}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{8}{9}}$ as $\frac{\sqrt{8}}{\sqrt{9}}$ .

$y = \pm \frac{\sqrt{8}}{\sqrt{9}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.2.1.1

Factor $4$ out of $8$ .

$y = \pm \frac{\sqrt{4 (2)}}{\sqrt{9}}$

Step 4.2.1.2

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

$y = \pm \frac{\sqrt{2^{2} \cdot 2}}{\sqrt{9}}$

Step 4.2.2

Pull terms out from under the radical.

$y = \pm \frac{2 \sqrt{2}}{\sqrt{9}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $9$ as $3^{2}$ .

$y = \pm \frac{2 \sqrt{2}}{\sqrt{3^{2}}}$

Step 4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm \frac{2 \sqrt{2}}{3}$

Step 5

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

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Step 5.1

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

$y = \frac{2 \sqrt{2}}{3}$

Step 5.2

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

$y = - \frac{2 \sqrt{2}}{3}$

Step 5.3

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

$y = \frac{2 \sqrt{2}}{3}, - \frac{2 \sqrt{2}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{2 \sqrt{2}}{3}, - \frac{2 \sqrt{2}}{3}$

Decimal Form:

$y = 0.94280904 \dots, - 0.94280904 \dots$