Algebra Examples

$\sqrt[7]{4^{- 10}}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt[7]{\frac{1}{4^{10}}}$

Step 2

Raise $4$ to the power of $10$ .

$\sqrt[7]{\frac{1}{1048576}}$

Step 3

Rewrite $\sqrt[7]{\frac{1}{1048576}}$ as $\frac{\sqrt[7]{1}}{\sqrt[7]{1048576}}$ .

$\frac{\sqrt[7]{1}}{\sqrt[7]{1048576}}$

Step 4

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[7]{1048576}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $1048576$ as $4^{7} \cdot 64$ .

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Step 5.1.1

Factor $16384$ out of $1048576$ .

$\frac{1}{\sqrt[7]{16384 (64)}}$

Step 5.1.2

Rewrite $16384$ as $4^{7}$ .

$\frac{1}{\sqrt[7]{4^{7} \cdot 64}}$

Step 5.2

Pull terms out from under the radical.

$\frac{1}{4 \sqrt[7]{64}}$

Step 6

Multiply $\frac{1}{4 \sqrt[7]{64}}$ by $\frac{{\sqrt[7]{64}}^{6}}{{\sqrt[7]{64}}^{6}}$ .

$\frac{1}{4 \sqrt[7]{64}} \cdot \frac{{\sqrt[7]{64}}^{6}}{{\sqrt[7]{64}}^{6}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{4 \sqrt[7]{64}}$ by $\frac{{\sqrt[7]{64}}^{6}}{{\sqrt[7]{64}}^{6}}$ .

$\frac{{\sqrt[7]{64}}^{6}}{4 \sqrt[7]{64} {\sqrt[7]{64}}^{6}}$

Step 7.2

Move $\sqrt[7]{64}$ .

$\frac{{\sqrt[7]{64}}^{6}}{4 (\sqrt[7]{64} {\sqrt[7]{64}}^{6})}$

Step 7.3

Raise $\sqrt[7]{64}$ to the power of $1$ .

$\frac{{\sqrt[7]{64}}^{6}}{4 ({\sqrt[7]{64}}^{1} {\sqrt[7]{64}}^{6})}$

Step 7.4

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

$\frac{{\sqrt[7]{64}}^{6}}{4 {\sqrt[7]{64}}^{1 + 6}}$

Step 7.5

Add $1$ and $6$ .

$\frac{{\sqrt[7]{64}}^{6}}{4 {\sqrt[7]{64}}^{7}}$

Step 7.6

Rewrite ${\sqrt[7]{64}}^{7}$ as $64$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{64}$ as $64^{\frac{1}{7}}$ .

$\frac{{\sqrt[7]{64}}^{6}}{4 {(64^{\frac{1}{7}})}^{7}}$

Step 7.6.2

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

$\frac{{\sqrt[7]{64}}^{6}}{4 \cdot 64^{\frac{1}{7} \cdot 7}}$

Step 7.6.3

Combine $\frac{1}{7}$ and $7$ .

$\frac{{\sqrt[7]{64}}^{6}}{4 \cdot 64^{\frac{7}{7}}}$

Step 7.6.4

Cancel the common factor of $7$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{{\sqrt[7]{64}}^{6}}{4 \cdot 64^{\frac{7}{7}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{{\sqrt[7]{64}}^{6}}{4 \cdot 64^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{{\sqrt[7]{64}}^{6}}{4 \cdot 64}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite ${\sqrt[7]{64}}^{6}$ as $\sqrt[7]{64^{6}}$ .

$\frac{\sqrt[7]{64^{6}}}{4 \cdot 64}$

Step 8.2

Raise $64$ to the power of $6$ .

$\frac{\sqrt[7]{68719476736}}{4 \cdot 64}$

Step 8.3

Rewrite $68719476736$ as $32^{7} \cdot 2$ .

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Step 8.3.1

Factor $34359738368$ out of $68719476736$ .

$\frac{\sqrt[7]{34359738368 (2)}}{4 \cdot 64}$

Step 8.3.2

Rewrite $34359738368$ as $32^{7}$ .

$\frac{\sqrt[7]{32^{7} \cdot 2}}{4 \cdot 64}$

Step 8.4

Pull terms out from under the radical.

$\frac{32 \sqrt[7]{2}}{4 \cdot 64}$

Step 9

Multiply $4$ by $64$ .

$\frac{32 \sqrt[7]{2}}{256}$

Step 10

Cancel the common factor of $32$ and $256$ .

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Step 10.1

Factor $32$ out of $32 \sqrt[7]{2}$ .

$\frac{32 (\sqrt[7]{2})}{256}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $32$ out of $256$ .

$\frac{32 \sqrt[7]{2}}{32 \cdot 8}$

Step 10.2.2

Cancel the common factor.

$\frac{32 \sqrt[7]{2}}{32 \cdot 8}$

Step 10.2.3

Rewrite the expression.

$\frac{\sqrt[7]{2}}{8}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[7]{2}}{8}$

Decimal Form:

$0.13801118 \dots$