Algebra Examples

$\frac{\sqrt[4]{2^{- 2}}}{\sqrt[6]{2^{- 9}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{\sqrt[4]{\frac{1}{2^{2}}}}{\sqrt[6]{2^{- 9}}}$

Step 1.2

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

$\frac{\sqrt[4]{\frac{1}{4}}}{\sqrt[6]{2^{- 9}}}$

Step 1.3

Rewrite $1$ as $1^{2}$ .

$\frac{\sqrt[4]{\frac{1^{2}}{4}}}{\sqrt[6]{2^{- 9}}}$

Step 1.4

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

$\frac{\sqrt[4]{\frac{1^{2}}{2^{2}}}}{\sqrt[6]{2^{- 9}}}$

Step 1.5

Rewrite $\frac{1^{2}}{2^{2}}$ as ${(\frac{1}{2})}^{2}$ .

$\frac{\sqrt[4]{{(\frac{1}{2})}^{2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.6

Rewrite $\sqrt[4]{{(\frac{1}{2})}^{2}}$ as $\sqrt{\sqrt{{(\frac{1}{2})}^{2}}}$ .

$\frac{\sqrt{\sqrt{{(\frac{1}{2})}^{2}}}}{\sqrt[6]{2^{- 9}}}$

Step 1.7

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

$\frac{\sqrt{\frac{1}{2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.8

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{1}}{\sqrt{2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.9

Any root of $1$ is $1$ .

$\frac{\frac{1}{\sqrt{2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.10

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11

Combine and simplify the denominator.

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Step 1.11.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.4

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

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.11.6.1

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

$\frac{\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.6.2

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

$\frac{\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{\sqrt{2}}{2^{\frac{2}{2}}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.6.4

Cancel the common factor of $2$ .

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Step 1.11.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{2}}{2^{\frac{2}{2}}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{2}}{2^{1}}}{\sqrt[6]{2^{- 9}}}$

Step 1.11.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{2}}{2}}{\sqrt[6]{2^{- 9}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{\sqrt{2}}{2}}{\sqrt[6]{\frac{1}{2^{9}}}}$

Step 2.2

Raise $2$ to the power of $9$ .

$\frac{\frac{\sqrt{2}}{2}}{\sqrt[6]{\frac{1}{512}}}$

Step 2.3

Rewrite $\sqrt[6]{\frac{1}{512}}$ as $\frac{\sqrt[6]{1}}{\sqrt[6]{512}}$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt[6]{1}}{\sqrt[6]{512}}}$

Step 2.4

Any root of $1$ is $1$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{1}{\sqrt[6]{512}}}$

Step 2.5

Simplify the denominator.

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Step 2.5.1

Rewrite $512$ as $2^{6} \cdot 8$ .

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Step 2.5.1.1

Factor $64$ out of $512$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{1}{\sqrt[6]{64 (8)}}}$

Step 2.5.1.2

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{1}{\sqrt[6]{2^{6} \cdot 8}}}$

Step 2.5.2

Pull terms out from under the radical.

$\frac{\frac{\sqrt{2}}{2}}{\frac{1}{2 \sqrt[6]{8}}}$

Step 2.5.3

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{1}{2 \sqrt[6]{2^{3}}}}$

Step 2.5.4

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{1}{2 \sqrt{\sqrt[3]{2^{3}}}}}$

Step 2.5.5

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{1}{2 \sqrt{2}}}$

Step 2.6

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}$

Step 2.7

Combine and simplify the denominator.

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Step 2.7.1

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}}}$

Step 2.7.2

Move $\sqrt{2}$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}}$

Step 2.7.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.7.4

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.7.5

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}}$

Step 2.7.6

Add $1$ and $1$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2 {\sqrt{2}}^{2}}}$

Step 2.7.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.7.7.1

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

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

Step 2.7.7.2

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}}$

Step 2.7.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}}$

Step 2.7.7.4

Cancel the common factor of $2$ .

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Step 2.7.7.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}}$

Step 2.7.7.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2 \cdot 2^{1}}}$

Step 2.7.7.5

Evaluate the exponent.

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2 \cdot 2}}$

Step 2.8

Multiply $2$ by $2$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{2}}{2} \cdot \frac{4}{\sqrt{2}}$

Step 4

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

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Step 4.1

Cancel the common factor.

$\frac{\sqrt{2}}{2} \cdot \frac{4}{\sqrt{2}}$

Step 4.2

Rewrite the expression.

$\frac{1}{2} \cdot 4$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Factor $2$ out of $4$ .

$\frac{1}{2} \cdot (2 (2))$

Step 5.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot 2)$

Step 5.3

Rewrite the expression.

$2$