Pre-Algebra Examples

$y = \frac{3}{2} x^{2}$

Step 1

Rewrite the equation as $\frac{3}{2} x^{2} = y$ .

$\frac{3}{2} x^{2} = y$

Step 2

Multiply both sides of the equation by $\frac{2}{3}$ .

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

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{2}{3} (\frac{3}{2} x^{2})$ .

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Step 3.1.1.1

Combine $\frac{3}{2}$ and $x^{2}$ .

$\frac{2}{3} \cdot \frac{3 x^{2}}{2} = \frac{2}{3} y$

Step 3.1.1.2

Combine.

$\frac{2 (3 x^{2})}{3 \cdot 2} = \frac{2}{3} y$

Step 3.1.1.3

Cancel the common factor of $2$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{2 (3 x^{2})}{3 \cdot 2} = \frac{2}{3} y$

Step 3.1.1.3.2

Rewrite the expression.

$\frac{3 x^{2}}{3} = \frac{2}{3} y$

Step 3.1.1.4

Cancel the common factor of $3$ .

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Step 3.1.1.4.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{2}{3} y$

Step 3.1.1.4.2

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

$x^{2} = \frac{2}{3} y$

Step 3.2

Simplify the right side.

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Step 3.2.1

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

$x^{2} = \frac{2 y}{3}$

Step 4

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

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

Step 5

Simplify $\pm \sqrt{\frac{2 y}{3}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{2 y}{3}}$ as $\frac{\sqrt{2 y}}{\sqrt{3}}$ .

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

Step 5.2

Multiply $\frac{\sqrt{2 y}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.3

Combine and simplify the denominator.

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Step 5.3.1

Multiply $\frac{\sqrt{2 y}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.3.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.3.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.3.4

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

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

Step 5.3.5

Add $1$ and $1$ .

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

Step 5.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 5.3.6.2

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

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

Step 5.3.6.3

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

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

Step 5.3.6.4

Cancel the common factor of $2$ .

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Step 5.3.6.4.1

Cancel the common factor.

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

Step 5.3.6.4.2

Rewrite the expression.

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

Step 5.3.6.5

Evaluate the exponent.

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

Step 5.4

Simplify the numerator.

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Step 5.4.1

Combine using the product rule for radicals.

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

Step 5.4.2

Multiply $3$ by $2$ .

$x = \pm \frac{\sqrt{6 y}}{3}$

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.

$x = \frac{\sqrt{6 y}}{3}$

Step 6.2

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

$x = - \frac{\sqrt{6 y}}{3}$

Step 6.3

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

$x = \frac{\sqrt{6 y}}{3}$

$x = - \frac{\sqrt{6 y}}{3}$

$x = \frac{\sqrt{6 y}}{3}$

$x = - \frac{\sqrt{6 y}}{3}$