Pre-Algebra Examples

$\sqrt[5]{8} \sqrt{8} \div \sqrt[3]{8^{5}}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt[5]{8} \sqrt{8}}{\sqrt[3]{8^{5}}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite the expression using the least common index of $10$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{8}$ as $8^{\frac{1}{5}}$ .

$\frac{8^{\frac{1}{5}} \sqrt{8}}{\sqrt[3]{8^{5}}}$

Step 2.1.2

Rewrite $8^{\frac{1}{5}}$ as $8^{\frac{2}{10}}$ .

$\frac{8^{\frac{2}{10}} \sqrt{8}}{\sqrt[3]{8^{5}}}$

Step 2.1.3

Rewrite $8^{\frac{2}{10}}$ as $\sqrt[10]{8^{2}}$ .

$\frac{\sqrt[10]{8^{2}} \sqrt{8}}{\sqrt[3]{8^{5}}}$

Step 2.1.4

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

$\frac{\sqrt[10]{8^{2}} \cdot 8^{\frac{1}{2}}}{\sqrt[3]{8^{5}}}$

Step 2.1.5

Rewrite $8^{\frac{1}{2}}$ as $8^{\frac{5}{10}}$ .

$\frac{\sqrt[10]{8^{2}} \cdot 8^{\frac{5}{10}}}{\sqrt[3]{8^{5}}}$

Step 2.1.6

Rewrite $8^{\frac{5}{10}}$ as $\sqrt[10]{8^{5}}$ .

$\frac{\sqrt[10]{8^{2}} \sqrt[10]{8^{5}}}{\sqrt[3]{8^{5}}}$

Step 2.2

Combine using the product rule for radicals.

$\frac{\sqrt[10]{8^{2} \cdot 8^{5}}}{\sqrt[3]{8^{5}}}$

Step 2.3

Multiply $8^{2}$ by $8^{5}$ by adding the exponents.

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Step 2.3.1

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

$\frac{\sqrt[10]{8^{2 + 5}}}{\sqrt[3]{8^{5}}}$

Step 2.3.2

Add $2$ and $5$ .

$\frac{\sqrt[10]{8^{7}}}{\sqrt[3]{8^{5}}}$

Step 3

Simplify the denominator.

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Step 3.1

Raise $8$ to the power of $5$ .

$\frac{\sqrt[10]{8^{7}}}{\sqrt[3]{32768}}$

Step 3.2

Rewrite $32768$ as $32^{3}$ .

$\frac{\sqrt[10]{8^{7}}}{\sqrt[3]{32^{3}}}$

Step 3.3

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

$\frac{\sqrt[10]{8^{7}}}{32}$

Step 4

Simplify the numerator.

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Step 4.1

Raise $8$ to the power of $7$ .

$\frac{\sqrt[10]{2097152}}{32}$

Step 4.2

Rewrite $2097152$ as $4^{10} \cdot 2$ .

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Step 4.2.1

Factor $1048576$ out of $2097152$ .

$\frac{\sqrt[10]{1048576 (2)}}{32}$

Step 4.2.2

Rewrite $1048576$ as $4^{10}$ .

$\frac{\sqrt[10]{4^{10} \cdot 2}}{32}$

Step 4.3

Pull terms out from under the radical.

$\frac{4 \sqrt[10]{2}}{32}$

Step 5

Cancel the common factor of $4$ and $32$ .

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Step 5.1

Factor $4$ out of $4 \sqrt[10]{2}$ .

$\frac{4 (\sqrt[10]{2})}{32}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $4$ out of $32$ .

$\frac{4 \sqrt[10]{2}}{4 \cdot 8}$

Step 5.2.2

Cancel the common factor.

$\frac{4 \sqrt[10]{2}}{4 \cdot 8}$

Step 5.2.3

Rewrite the expression.

$\frac{\sqrt[10]{2}}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[10]{2}}{8}$

Decimal Form:

$0.13397168 \dots$