Algebra Examples

$\sqrt[4]{8} \cdot \sqrt[3]{32}$

Step 1

Rewrite $32$ as $2^{3} \cdot 4$ .

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Step 1.1

Factor $8$ out of $32$ .

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

Step 1.2

Rewrite $8$ as $2^{3}$ .

$\sqrt[4]{8} \cdot \sqrt[3]{2^{3} \cdot 4}$

Step 2

Pull terms out from under the radical.

$\sqrt[4]{8} \cdot (2 \sqrt[3]{4})$

Step 3

Multiply $\sqrt[4]{8} (2 \sqrt[3]{4})$ .

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Step 3.1

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

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$2 (8^{\frac{1}{4}} \sqrt[3]{4})$

Step 3.1.2

Rewrite $8^{\frac{1}{4}}$ as $8^{\frac{3}{12}}$ .

$2 (8^{\frac{3}{12}} \sqrt[3]{4})$

Step 3.1.3

Rewrite $8^{\frac{3}{12}}$ as $\sqrt[12]{8^{3}}$ .

$2 (\sqrt[12]{8^{3}} \sqrt[3]{4})$

Step 3.1.4

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

$2 (\sqrt[12]{8^{3}} \cdot 4^{\frac{1}{3}})$

Step 3.1.5

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

$2 (\sqrt[12]{8^{3}} \cdot 4^{\frac{4}{12}})$

Step 3.1.6

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

$2 (\sqrt[12]{8^{3}} \sqrt[12]{4^{4}})$

Step 3.2

Combine using the product rule for radicals.

$2 \sqrt[12]{8^{3} \cdot 4^{4}}$

Step 3.3

Rewrite $8$ as $2^{3}$ .

$2 \sqrt[12]{{(2^{3})}^{3} \cdot 4^{4}}$

Step 3.4

Multiply the exponents in ${(2^{3})}^{3}$ .

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Step 3.4.1

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

$2 \sqrt[12]{2^{3 \cdot 3} \cdot 4^{4}}$

Step 3.4.2

Multiply $3$ by $3$ .

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

Step 3.5

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

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

Step 3.6

Multiply the exponents in ${(2^{2})}^{4}$ .

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Step 3.6.1

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

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

Step 3.6.2

Multiply $2$ by $4$ .

$2 \sqrt[12]{2^{9} \cdot 2^{8}}$

Step 3.7

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

$2 \sqrt[12]{2^{9 + 8}}$

Step 3.8

Add $9$ and $8$ .

$2 \sqrt[12]{2^{17}}$

Step 4

Raise $2$ to the power of $17$ .

$2 \sqrt[12]{131072}$

Step 5

Rewrite $131072$ as $2^{12} \cdot 32$ .

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Step 5.1

Factor $4096$ out of $131072$ .

$2 \sqrt[12]{4096 (32)}$

Step 5.2

Rewrite $4096$ as $2^{12}$ .

$2 \sqrt[12]{2^{12} \cdot 32}$

Step 6

Pull terms out from under the radical.

$2 (2 \sqrt[12]{32})$

Step 7

Multiply $2$ by $2$ .

$4 \sqrt[12]{32}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt[12]{32}$

Decimal Form:

$5.33935941 \dots$