Algebra Examples

$\frac{\sqrt[3]{x^{2}} \cdot \sqrt{x^{5}}}{\sqrt[6]{x}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the least common index of $6$ .

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Step 1.1.1

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

$\frac{x^{\frac{2}{3}} \sqrt{x^{5}}}{\sqrt[6]{x}}$

Step 1.1.2

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{4}{6}}$ .

$\frac{x^{\frac{4}{6}} \sqrt{x^{5}}}{\sqrt[6]{x}}$

Step 1.1.3

Rewrite $x^{\frac{4}{6}}$ as $\sqrt[6]{x^{4}}$ .

$\frac{\sqrt[6]{x^{4}} \sqrt{x^{5}}}{\sqrt[6]{x}}$

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5}}$ as $x^{\frac{5}{2}}$ .

$\frac{\sqrt[6]{x^{4}} x^{\frac{5}{2}}}{\sqrt[6]{x}}$

Step 1.1.5

Rewrite $x^{\frac{5}{2}}$ as $x^{\frac{15}{6}}$ .

$\frac{\sqrt[6]{x^{4}} x^{\frac{15}{6}}}{\sqrt[6]{x}}$

Step 1.1.6

Rewrite $x^{\frac{15}{6}}$ as $\sqrt[6]{x^{15}}$ .

$\frac{\sqrt[6]{x^{4}} \sqrt[6]{x^{15}}}{\sqrt[6]{x}}$

Step 1.2

Combine using the product rule for radicals.

$\frac{\sqrt[6]{x^{4} x^{15}}}{\sqrt[6]{x}}$

Step 1.3

Multiply $x^{4}$ by $x^{15}$ by adding the exponents.

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Step 1.3.1

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

$\frac{\sqrt[6]{x^{4 + 15}}}{\sqrt[6]{x}}$

Step 1.3.2

Add $4$ and $15$ .

$\frac{\sqrt[6]{x^{19}}}{\sqrt[6]{x}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $x^{19}$ as ${(x^{3})}^{6} x$ .

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Step 2.1.1

Factor out $x^{18}$ .

$\frac{\sqrt[6]{x^{18} x}}{\sqrt[6]{x}}$

Step 2.1.2

Rewrite $x^{18}$ as ${(x^{3})}^{6}$ .

$\frac{\sqrt[6]{{(x^{3})}^{6} x}}{\sqrt[6]{x}}$

Step 2.2

Pull terms out from under the radical.

$\frac{x^{3} \sqrt[6]{x}}{\sqrt[6]{x}}$

Step 3

Cancel the common factor of $\sqrt[6]{x}$ .

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Step 3.1

Cancel the common factor.

$\frac{x^{3} \sqrt[6]{x}}{\sqrt[6]{x}}$

Step 3.2

Divide $x^{3}$ by $1$ .

$x^{3}$