Algebra Examples

$\sqrt[3]{4} \cdot \sqrt{8}$

Step 1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1

Factor $4$ out of $8$ .

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

Step 1.2

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$2 (4^{\frac{1}{3}} \sqrt{2})$

Step 3.1.2

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{2}{6}}$ .

$2 (4^{\frac{2}{6}} \sqrt{2})$

Step 3.1.3

Rewrite $4^{\frac{2}{6}}$ as $\sqrt[6]{4^{2}}$ .

$2 (\sqrt[6]{4^{2}} \sqrt{2})$

Step 3.1.4

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

$2 (\sqrt[6]{4^{2}} \cdot 2^{\frac{1}{2}})$

Step 3.1.5

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

$2 (\sqrt[6]{4^{2}} \cdot 2^{\frac{3}{6}})$

Step 3.1.6

Rewrite $2^{\frac{3}{6}}$ as $\sqrt[6]{2^{3}}$ .

$2 (\sqrt[6]{4^{2}} \sqrt[6]{2^{3}})$

Step 3.2

Combine using the product rule for radicals.

$2 \sqrt[6]{4^{2} \cdot 2^{3}}$

Step 3.3

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

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

Step 3.4

Multiply the exponents in ${(2^{2})}^{2}$ .

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Step 3.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 \sqrt[6]{2^{2 \cdot 2} \cdot 2^{3}}$

Step 3.4.2

Multiply $2$ by $2$ .

$2 \sqrt[6]{2^{4} \cdot 2^{3}}$

Step 3.5

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

$2 \sqrt[6]{2^{4 + 3}}$

Step 3.6

Add $4$ and $3$ .

$2 \sqrt[6]{2^{7}}$

Step 4

Raise $2$ to the power of $7$ .

$2 \sqrt[6]{128}$

Step 5

Rewrite $128$ as $2^{6} \cdot 2$ .

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Step 5.1

Factor $64$ out of $128$ .

$2 \sqrt[6]{64 (2)}$

Step 5.2

Rewrite $64$ as $2^{6}$ .

$2 \sqrt[6]{2^{6} \cdot 2}$

Step 6

Pull terms out from under the radical.

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

Step 7

Multiply $2$ by $2$ .

$4 \sqrt[6]{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt[6]{2}$

Decimal Form:

$4.48984819 \dots$