Solve for y (-( square root of 8)/3)^2+y^2=1^2

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

Step 1

Simplify each term.

Simplify the numerator.

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

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the numerator.

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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.

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

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

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

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

Combine the numerators over the common denominator.

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

Subtract $8$ from $9$ .

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

Step 4

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

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

Step 5

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

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

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

Any root of $1$ is $1$ .

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

Simplify the denominator.

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

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

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.

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

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

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

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

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:

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

Solve for y (-( square root of 8)/3)^2+y^2=1^2

Related Questions

Support LCMGCF.com by disabling your adblocker.