Algebra Examples

$\sqrt[6]{x} \cdot \sqrt[5]{x} \cdot \sqrt[10]{x^{3}}$

Step 1

Multiply $\sqrt[6]{x} \sqrt[5]{x}$ .

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Step 1.1

Rewrite the expression using the least common index of $30$ .

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$x^{\frac{1}{6}} \sqrt[5]{x} \cdot \sqrt[10]{x^{3}}$

Step 1.1.2

Rewrite $x^{\frac{1}{6}}$ as $x^{\frac{5}{30}}$ .

$x^{\frac{5}{30}} \sqrt[5]{x} \cdot \sqrt[10]{x^{3}}$

Step 1.1.3

Rewrite $x^{\frac{5}{30}}$ as $\sqrt[30]{x^{5}}$ .

$\sqrt[30]{x^{5}} \sqrt[5]{x} \cdot \sqrt[10]{x^{3}}$

Step 1.1.4

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

$\sqrt[30]{x^{5}} x^{\frac{1}{5}} \cdot \sqrt[10]{x^{3}}$

Step 1.1.5

Rewrite $x^{\frac{1}{5}}$ as $x^{\frac{6}{30}}$ .

$\sqrt[30]{x^{5}} x^{\frac{6}{30}} \cdot \sqrt[10]{x^{3}}$

Step 1.1.6

Rewrite $x^{\frac{6}{30}}$ as $\sqrt[30]{x^{6}}$ .

$\sqrt[30]{x^{5}} \sqrt[30]{x^{6}} \cdot \sqrt[10]{x^{3}}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt[30]{x^{5} x^{6}} \cdot \sqrt[10]{x^{3}}$

Step 1.3

Multiply $x^{5}$ by $x^{6}$ 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.

$\sqrt[30]{x^{5 + 6}} \cdot \sqrt[10]{x^{3}}$

Step 1.3.2

Add $5$ and $6$ .

$\sqrt[30]{x^{11}} \cdot \sqrt[10]{x^{3}}$

Step 2

Multiply $\sqrt[30]{x^{11}} \sqrt[10]{x^{3}}$ .

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Step 2.1

Rewrite the expression using the least common index of $30$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[10]{x^{3}}$ as $x^{\frac{3}{10}}$ .

$\sqrt[30]{x^{11}} x^{\frac{3}{10}}$

Step 2.1.2

Rewrite $x^{\frac{3}{10}}$ as $x^{\frac{9}{30}}$ .

$\sqrt[30]{x^{11}} x^{\frac{9}{30}}$

Step 2.1.3

Rewrite $x^{\frac{9}{30}}$ as $\sqrt[30]{x^{9}}$ .

$\sqrt[30]{x^{11}} \sqrt[30]{x^{9}}$

Step 2.2

Combine using the product rule for radicals.

$\sqrt[30]{x^{11} x^{9}}$

Step 2.3

Multiply $x^{11}$ by $x^{9}$ by adding the exponents.

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Step 2.3.1

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

$\sqrt[30]{x^{11 + 9}}$

Step 2.3.2

Add $11$ and $9$ .

$\sqrt[30]{x^{20}}$

Step 3

Rewrite $x^{20}$ as ${(x^{4})}^{5}$ .

$\sqrt[30]{{(x^{4})}^{5}}$

Step 4

Rewrite $\sqrt[30]{{(x^{4})}^{5}}$ as $\sqrt[6]{\sqrt[5]{{(x^{4})}^{5}}}$ .

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

Step 5

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

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

Step 6

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\sqrt[6]{{(x^{2})}^{2}}$

Step 7

Rewrite $\sqrt[6]{{(x^{2})}^{2}}$ as $\sqrt[3]{\sqrt{{(x^{2})}^{2}}}$ .

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

Step 8

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

$\sqrt[3]{x^{2}}$