Precalculus Examples

$\frac{1}{\sqrt[4]{x^{2} - 5 x}}$

Step 1

Factor $x$ out of $x^{2} - 5 x$ .

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Step 1.1

Factor $x$ out of $x^{2}$ .

$\frac{1}{\sqrt[4]{x \cdot x - 5 x}}$

Step 1.2

Factor $x$ out of $- 5 x$ .

$\frac{1}{\sqrt[4]{x \cdot x + x \cdot - 5}}$

Step 1.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$\frac{1}{\sqrt[4]{x (x - 5)}}$

Step 2

Multiply $\frac{1}{\sqrt[4]{x (x - 5)}}$ by $\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{\sqrt[4]{x (x - 5)}}^{3}}$ .

$\frac{1}{\sqrt[4]{x (x - 5)}} \cdot \frac{{\sqrt[4]{x (x - 5)}}^{3}}{{\sqrt[4]{x (x - 5)}}^{3}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{1}{\sqrt[4]{x (x - 5)}}$ by $\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{\sqrt[4]{x (x - 5)}}^{3}}$ .

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{\sqrt[4]{x (x - 5)} {\sqrt[4]{x (x - 5)}}^{3}}$

Step 3.2

Raise $\sqrt[4]{x (x - 5)}$ to the power of $1$ .

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{\sqrt[4]{x (x - 5)}}^{1} {\sqrt[4]{x (x - 5)}}^{3}}$

Step 3.3

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

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{\sqrt[4]{x (x - 5)}}^{1 + 3}}$

Step 3.4

Add $1$ and $3$ .

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{\sqrt[4]{x (x - 5)}}^{4}}$

Step 3.5

Rewrite ${\sqrt[4]{x (x - 5)}}^{4}$ as $x (x - 5)$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x (x - 5)}$ as ${(x (x - 5))}^{\frac{1}{4}}$ .

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{({(x (x - 5))}^{\frac{1}{4}})}^{4}}$

Step 3.5.2

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

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{(x (x - 5))}^{\frac{1}{4} \cdot 4}}$

Step 3.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{(x (x - 5))}^{\frac{4}{4}}}$

Step 3.5.4

Cancel the common factor of $4$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{(x (x - 5))}^{\frac{4}{4}}}$

Step 3.5.4.2

Rewrite the expression.

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{{(x (x - 5))}^{1}}$

Step 3.5.5

Simplify.

$\frac{{\sqrt[4]{x (x - 5)}}^{3}}{x (x - 5)}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[4]{x (x - 5)}}^{3}$ as $\sqrt[4]{{(x (x - 5))}^{3}}$ .

$\frac{\sqrt[4]{{(x (x - 5))}^{3}}}{x (x - 5)}$

Step 4.2

Apply the product rule to $x (x - 5)$ .

$\frac{\sqrt[4]{x^{3} {(x - 5)}^{3}}}{x (x - 5)}$