Algebra Examples

$\sqrt[3]{125 x^{8}} \cdot \sqrt[4]{16 x}$

Step 1

Rewrite $125 x^{8}$ as ${(5 x^{2})}^{3} x^{2}$ .

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Step 1.1

Rewrite $125$ as $5^{3}$ .

$\sqrt[3]{5^{3} x^{8}} \cdot \sqrt[4]{16 x}$

Step 1.2

Factor out $x^{6}$ .

$\sqrt[3]{5^{3} (x^{6} x^{2})} \cdot \sqrt[4]{16 x}$

Step 1.3

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\sqrt[3]{5^{3} ({(x^{2})}^{3} x^{2})} \cdot \sqrt[4]{16 x}$

Step 1.4

Rewrite $5^{3} {(x^{2})}^{3}$ as ${(5 x^{2})}^{3}$ .

$\sqrt[3]{{(5 x^{2})}^{3} x^{2}} \cdot \sqrt[4]{16 x}$

Step 2

Pull terms out from under the radical.

$5 x^{2} \sqrt[3]{x^{2}} \cdot \sqrt[4]{16 x}$

Step 3

Rewrite $16$ as $2^{4}$ .

$5 x^{2} \sqrt[3]{x^{2}} \cdot \sqrt[4]{2^{4} x}$

Step 4

Pull terms out from under the radical.

$5 x^{2} \sqrt[3]{x^{2}} \cdot (| 2 | \sqrt[4]{x})$

Step 5

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$5 x^{2} \sqrt[3]{x^{2}} \cdot (2 \sqrt[4]{x})$

Step 6

Multiply $5 x^{2} \sqrt[3]{x^{2}} (2 \sqrt[4]{x})$ .

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Step 6.1

Multiply $2$ by $5$ .

$10 x^{2} \sqrt[3]{x^{2}} \sqrt[4]{x}$

Step 6.2

Rewrite the expression using the least common index of $12$ .

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Step 6.2.1

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

$10 x^{2} (x^{\frac{1}{4}} \sqrt[3]{x^{2}})$

Step 6.2.2

Rewrite $x^{\frac{1}{4}}$ as $x^{\frac{3}{12}}$ .

$10 x^{2} (x^{\frac{3}{12}} \sqrt[3]{x^{2}})$

Step 6.2.3

Rewrite $x^{\frac{3}{12}}$ as $\sqrt[12]{x^{3}}$ .

$10 x^{2} (\sqrt[12]{x^{3}} \sqrt[3]{x^{2}})$

Step 6.2.4

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

$10 x^{2} (\sqrt[12]{x^{3}} x^{\frac{2}{3}})$

Step 6.2.5

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{8}{12}}$ .

$10 x^{2} (\sqrt[12]{x^{3}} x^{\frac{8}{12}})$

Step 6.2.6

Rewrite $x^{\frac{8}{12}}$ as $\sqrt[12]{x^{8}}$ .

$10 x^{2} (\sqrt[12]{x^{3}} \sqrt[12]{x^{8}})$

Step 6.3

Combine using the product rule for radicals.

$10 x^{2} \sqrt[12]{x^{3} x^{8}}$

Step 6.4

Multiply $x^{3}$ by $x^{8}$ by adding the exponents.

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Step 6.4.1

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

$10 x^{2} \sqrt[12]{x^{3 + 8}}$

Step 6.4.2

Add $3$ and $8$ .

$10 x^{2} \sqrt[12]{x^{11}}$