Algebra Examples

${(- \frac{\sqrt{8}}{3})}^{2} + y^{2} = 1^{2}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

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

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Step 1.1.1.1

Factor $4$ out of $8$ .

${(- \frac{\sqrt{4 (2)}}{3})}^{2} + y^{2} = 1^{2}$

Step 1.1.1.2

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

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

Step 1.1.2

Pull terms out from under the radical.

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

Step 1.2

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

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Step 1.2.1

Apply the product rule to $- \frac{2 \sqrt{2}}{3}$ .

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

Step 1.2.2

Apply the product rule to $\frac{2 \sqrt{2}}{3}$ .

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

Step 1.2.3

Apply the product rule to $2 \sqrt{2}$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

$\frac{4 {\sqrt{2}}^{2}}{3^{2}} + y^{2} = 1^{2}$

Step 1.5.2

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

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Step 1.5.2.1

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

$\frac{4 {(2^{\frac{1}{2}})}^{2}}{3^{2}} + y^{2} = 1^{2}$

Step 1.5.2.2

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

$\frac{4 \cdot 2^{\frac{1}{2} \cdot 2}}{3^{2}} + y^{2} = 1^{2}$

Step 1.5.2.3

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

$\frac{4 \cdot 2^{\frac{2}{2}}}{3^{2}} + y^{2} = 1^{2}$

Step 1.5.2.4

Cancel the common factor of $2$ .

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Step 1.5.2.4.1

Cancel the common factor.

$\frac{4 \cdot 2^{\frac{2}{2}}}{3^{2}} + y^{2} = 1^{2}$

Step 1.5.2.4.2

Rewrite the expression.

$\frac{4 \cdot 2^{1}}{3^{2}} + y^{2} = 1^{2}$

Step 1.5.2.5

Evaluate the exponent.

$\frac{4 \cdot 2}{3^{2}} + y^{2} = 1^{2}$

Step 1.6

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

$\frac{4 \cdot 2}{9} + y^{2} = 1^{2}$

Step 1.7

Multiply $4$ by $2$ .

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

Step 2

One to any power is one.

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

Step 3

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

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Step 3.1

Subtract $\frac{8}{9}$ from both sides of the equation.

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Subtract $8$ from $9$ .

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

Step 4

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

$y = \pm \sqrt{\frac{1}{9}}$

Step 5

Simplify $\pm \sqrt{\frac{1}{9}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{1}{9}}$ as $\frac{\sqrt{1}}{\sqrt{9}}$ .

$y = \pm \frac{\sqrt{1}}{\sqrt{9}}$

Step 5.2

Any root of $1$ is $1$ .

$y = \pm \frac{1}{\sqrt{9}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

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

$y = \pm \frac{1}{\sqrt{3^{2}}}$

Step 5.3.2

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

$y = \pm \frac{1}{3}$

Step 6

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

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Step 6.1

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

$y = \frac{1}{3}$

Step 6.2

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

$y = - \frac{1}{3}$

Step 6.3

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

$y = \frac{1}{3}, - \frac{1}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \frac{1}{3}, - \frac{1}{3}$

Decimal Form:

$y = 0.\overline{3}, - 0.\overline{3}$