Algebra Examples

$\sqrt[3]{125 x^{7}} \cdot \sqrt[4]{256 x^{3}}$

Step 1

Rewrite $125 x^{7}$ as ${(5 x^{2})}^{3} x$ .

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Step 1.1

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

$\sqrt[3]{5^{3} x^{7}} \cdot \sqrt[4]{256 x^{3}}$

Step 1.2

Factor out $x^{6}$ .

$\sqrt[3]{5^{3} (x^{6} x)} \cdot \sqrt[4]{256 x^{3}}$

Step 1.3

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

$\sqrt[3]{5^{3} ({(x^{2})}^{3} x)} \cdot \sqrt[4]{256 x^{3}}$

Step 1.4

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

$\sqrt[3]{{(5 x^{2})}^{3} x} \cdot \sqrt[4]{256 x^{3}}$

Step 2

Pull terms out from under the radical.

$5 x^{2} \sqrt[3]{x} \cdot \sqrt[4]{256 x^{3}}$

Step 3

Rewrite $256$ as $4^{4}$ .

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Multiply $4$ by $5$ .

$20 x^{2} \sqrt[3]{x} \sqrt[4]{x^{3}}$

Step 5.2

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

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Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

$20 x^{2} (x^{\frac{3}{4}} \sqrt[3]{x})$

Step 5.2.2

Rewrite $x^{\frac{3}{4}}$ as $x^{\frac{9}{12}}$ .

$20 x^{2} (x^{\frac{9}{12}} \sqrt[3]{x})$

Step 5.2.3

Rewrite $x^{\frac{9}{12}}$ as $\sqrt[12]{x^{9}}$ .

$20 x^{2} (\sqrt[12]{x^{9}} \sqrt[3]{x})$

Step 5.2.4

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

$20 x^{2} (\sqrt[12]{x^{9}} x^{\frac{1}{3}})$

Step 5.2.5

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

$20 x^{2} (\sqrt[12]{x^{9}} x^{\frac{4}{12}})$

Step 5.2.6

Rewrite $x^{\frac{4}{12}}$ as $\sqrt[12]{x^{4}}$ .

$20 x^{2} (\sqrt[12]{x^{9}} \sqrt[12]{x^{4}})$

Step 5.3

Combine using the product rule for radicals.

$20 x^{2} \sqrt[12]{x^{9} x^{4}}$

Step 5.4

Multiply $x^{9}$ by $x^{4}$ by adding the exponents.

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Step 5.4.1

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

$20 x^{2} \sqrt[12]{x^{9 + 4}}$

Step 5.4.2

Add $9$ and $4$ .

$20 x^{2} \sqrt[12]{x^{13}}$

Step 6

Factor out $x^{12}$ .

$20 x^{2} \sqrt[12]{x^{12} x}$

Step 7

Pull terms out from under the radical.

$20 x^{2} (x \sqrt[12]{x})$

Step 8

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.1

Move $x$ .

$20 (x \cdot x^{2}) \sqrt[12]{x}$

Step 8.2

Multiply $x$ by $x^{2}$ .

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Step 8.2.1

Raise $x$ to the power of $1$ .

$20 (x^{1} x^{2}) \sqrt[12]{x}$

Step 8.2.2

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

$20 x^{1 + 2} \sqrt[12]{x}$

Step 8.3

Add $1$ and $2$ .

$20 x^{3} \sqrt[12]{x}$