Algebra Examples

$\sqrt[4]{256 x} \cdot \sqrt[3]{125 x^{2}}$

Step 1

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

$\sqrt[4]{4^{4} x} \cdot \sqrt[3]{125 x^{2}}$

Step 2

Pull terms out from under the radical.

$4 \sqrt[4]{x} \cdot \sqrt[3]{125 x^{2}}$

Step 3

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Multiply $5$ by $4$ .

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

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 $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

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

Step 5.2.2

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

$20 (x^{\frac{8}{12}} \sqrt[4]{x})$

Step 5.2.3

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

$20 (\sqrt[12]{x^{8}} \sqrt[4]{x})$

Step 5.2.4

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

$20 (\sqrt[12]{x^{8}} x^{\frac{1}{4}})$

Step 5.2.5

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

$20 (\sqrt[12]{x^{8}} x^{\frac{3}{12}})$

Step 5.2.6

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

$20 (\sqrt[12]{x^{8}} \sqrt[12]{x^{3}})$

Step 5.3

Combine using the product rule for radicals.

$20 \sqrt[12]{x^{8} x^{3}}$

Step 5.4

Multiply $x^{8}$ by $x^{3}$ 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.

$20 \sqrt[12]{x^{8 + 3}}$

Step 5.4.2

Add $8$ and $3$ .

$20 \sqrt[12]{x^{11}}$