Algebra Examples

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

Step 1

Rewrite $16 x$ as ${(2^{2})}^{2} x$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Move $2^{2}$ .

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

Step 3.2

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

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Step 3.2.1

Raise $2$ to the power of $1$ .

$2^{2} \cdot 2^{1} \sqrt[3]{x^{2}} \cdot \sqrt{x}$

Step 3.2.2

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

$2^{2 + 1} \sqrt[3]{x^{2}} \cdot \sqrt{x}$

Step 3.3

Add $2$ and $1$ .

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

Step 4

Raise $2$ to the power of $3$ .

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

Step 5

Multiply $8 \sqrt[3]{x^{2}} \sqrt{x}$ .

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Step 5.1

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

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$8 (x^{\frac{1}{2}} \sqrt[3]{x^{2}})$

Step 5.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

$8 (x^{\frac{3}{6}} \sqrt[3]{x^{2}})$

Step 5.1.3

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

$8 (\sqrt[6]{x^{3}} \sqrt[3]{x^{2}})$

Step 5.1.4

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

$8 (\sqrt[6]{x^{3}} x^{\frac{2}{3}})$

Step 5.1.5

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

$8 (\sqrt[6]{x^{3}} x^{\frac{4}{6}})$

Step 5.1.6

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

$8 (\sqrt[6]{x^{3}} \sqrt[6]{x^{4}})$

Step 5.2

Combine using the product rule for radicals.

$8 \sqrt[6]{x^{3} x^{4}}$

Step 5.3

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 5.3.1

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

$8 \sqrt[6]{x^{3 + 4}}$

Step 5.3.2

Add $3$ and $4$ .

$8 \sqrt[6]{x^{7}}$

Step 6

Factor out $x^{6}$ .

$8 \sqrt[6]{x^{6} x}$

Step 7

Pull terms out from under the radical.

$8 x \sqrt[6]{x}$