Algebra Examples

$\sqrt{8} \cdot \sqrt[4]{8} \div \sqrt[12]{8}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt{8} \cdot \sqrt[4]{8}}{\sqrt[12]{8}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite the expression using the least common index of $4$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8}$ as $8^{\frac{1}{2}}$ .

$\frac{8^{\frac{1}{2}} \sqrt[4]{8}}{\sqrt[12]{8}}$

Step 2.1.2

Rewrite $8^{\frac{1}{2}}$ as $8^{\frac{2}{4}}$ .

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

Step 2.1.3

Rewrite $8^{\frac{2}{4}}$ as $\sqrt[4]{8^{2}}$ .

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

Step 2.2

Combine using the product rule for radicals.

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

Step 2.3

Multiply $8^{2}$ by $8$ by adding the exponents.

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Step 2.3.1

Multiply $8^{2}$ by $8$ .

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Step 2.3.1.1

Raise $8$ to the power of $1$ .

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

Step 2.3.1.2

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

$\frac{\sqrt[4]{8^{2 + 1}}}{\sqrt[12]{8}}$

Step 2.3.2

Add $2$ and $1$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

Rewrite $\sqrt[12]{2^{3}}$ as $\sqrt[4]{\sqrt[3]{2^{3}}}$ .

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

Step 3.3

Pull terms out from under the radical, assuming real numbers.

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

Step 4

Simplify the numerator.

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Step 4.1

Raise $8$ to the power of $3$ .

$\frac{\sqrt[4]{512}}{\sqrt[4]{2}}$

Step 4.2

Rewrite $512$ as $4^{4} \cdot 2$ .

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Step 4.2.1

Factor $256$ out of $512$ .

$\frac{\sqrt[4]{256 (2)}}{\sqrt[4]{2}}$

Step 4.2.2

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

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

Step 4.3

Pull terms out from under the radical.

$\frac{4 \sqrt[4]{2}}{\sqrt[4]{2}}$

Step 5

Cancel the common factor of $\sqrt[4]{2}$ .

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Step 5.1

Cancel the common factor.

$\frac{4 \sqrt[4]{2}}{\sqrt[4]{2}}$

Step 5.2

Divide $4$ by $1$ .

$4$