Algebra Examples

$\frac{1}{3} y^{2} - \frac{8}{27} = 0$

Step 1

Combine $\frac{1}{3}$ and $y^{2}$ .

$\frac{y^{2}}{3} - \frac{8}{27} = 0$

Step 2

Add $\frac{8}{27}$ to both sides of the equation.

$\frac{y^{2}}{3} = \frac{8}{27}$

Step 3

Multiply both sides of the equation by $3$ .

$3 \frac{y^{2}}{3} = 3 (\frac{8}{27})$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1

Cancel the common factor.

$3 \frac{y^{2}}{3} = 3 (\frac{8}{27})$

Step 4.1.1.2

Rewrite the expression.

$y^{2} = 3 (\frac{8}{27})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Factor $3$ out of $27$ .

$y^{2} = 3 \frac{8}{3 (9)}$

Step 4.2.1.2

Cancel the common factor.

$y^{2} = 3 \frac{8}{3 \cdot 9}$

Step 4.2.1.3

Rewrite the expression.

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

Step 5

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 6

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

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Step 6.1

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

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

Step 6.2

Simplify the numerator.

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Step 6.2.1

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

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Step 6.2.1.1

Factor $4$ out of $8$ .

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

Step 6.2.1.2

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

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

Step 6.2.2

Pull terms out from under the radical.

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

Step 6.3

Simplify the denominator.

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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