Pre-Algebra Examples

$\frac{1}{\sqrt[3]{x^{2} - 4}}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{1}{\sqrt[3]{x^{2} - 2^{2}}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{1}{\sqrt[3]{(x + 2) (x - 2)}}$

Step 2

Multiply $\frac{1}{\sqrt[3]{(x + 2) (x - 2)}}$ by $\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}$ .

$\frac{1}{\sqrt[3]{(x + 2) (x - 2)}} \cdot \frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{1}{\sqrt[3]{(x + 2) (x - 2)}}$ by $\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}$ .

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{\sqrt[3]{(x + 2) (x - 2)} {\sqrt[3]{(x + 2) (x - 2)}}^{2}}$

Step 3.2

Raise $\sqrt[3]{(x + 2) (x - 2)}$ to the power of $1$ .

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{\sqrt[3]{(x + 2) (x - 2)}}^{1} {\sqrt[3]{(x + 2) (x - 2)}}^{2}}$

Step 3.3

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

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{\sqrt[3]{(x + 2) (x - 2)}}^{1 + 2}}$

Step 3.4

Add $1$ and $2$ .

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{\sqrt[3]{(x + 2) (x - 2)}}^{3}}$

Step 3.5

Rewrite ${\sqrt[3]{(x + 2) (x - 2)}}^{3}$ as $(x + 2) (x - 2)$ .

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Step 3.5.1

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

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{({((x + 2) (x - 2))}^{\frac{1}{3}})}^{3}}$

Step 3.5.2

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

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{((x + 2) (x - 2))}^{\frac{1}{3} \cdot 3}}$

Step 3.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{((x + 2) (x - 2))}^{\frac{3}{3}}}$

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{((x + 2) (x - 2))}^{\frac{3}{3}}}$

Step 3.5.4.2

Rewrite the expression.

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{{((x + 2) (x - 2))}^{1}}$

Step 3.5.5

Simplify.

$\frac{{\sqrt[3]{(x + 2) (x - 2)}}^{2}}{(x + 2) (x - 2)}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[3]{(x + 2) (x - 2)}}^{2}$ as $\sqrt[3]{{((x + 2) (x - 2))}^{2}}$ .

$\frac{\sqrt[3]{{((x + 2) (x - 2))}^{2}}}{(x + 2) (x - 2)}$

Step 4.2

Apply the product rule to $(x + 2) (x - 2)$ .

$\frac{\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}{(x + 2) (x - 2)}$