Finite Math Examples

${(\frac{1}{\sqrt{32}})}^{- \frac{2}{5}}$

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $16$ out of $32$ .

${(\frac{1}{\sqrt{16 (2)}})}^{- \frac{2}{5}}$

Step 1.1.2

Rewrite $16$ as $4^{2}$ .

${(\frac{1}{\sqrt{4^{2} \cdot 2}})}^{- \frac{2}{5}}$

Step 1.2

Pull terms out from under the radical.

${(\frac{1}{4 \sqrt{2}})}^{- \frac{2}{5}}$

Step 2

Multiply $\frac{1}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{1}{4 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{- \frac{2}{5}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{1}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{\sqrt{2}}{4 \sqrt{2} \sqrt{2}})}^{- \frac{2}{5}}$

Step 3.2

Move $\sqrt{2}$ .

${(\frac{\sqrt{2}}{4 (\sqrt{2} \sqrt{2})})}^{- \frac{2}{5}}$

Step 3.3

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

${(\frac{\sqrt{2}}{4 ({\sqrt{2}}^{1} \sqrt{2})})}^{- \frac{2}{5}}$

Step 3.4

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

${(\frac{\sqrt{2}}{4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})})}^{- \frac{2}{5}}$

Step 3.5

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

${(\frac{\sqrt{2}}{4 {\sqrt{2}}^{1 + 1}})}^{- \frac{2}{5}}$

Step 3.6

Add $1$ and $1$ .

${(\frac{\sqrt{2}}{4 {\sqrt{2}}^{2}})}^{- \frac{2}{5}}$

Step 3.7

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

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Step 3.7.1

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

${(\frac{\sqrt{2}}{4 {(2^{\frac{1}{2}})}^{2}})}^{- \frac{2}{5}}$

Step 3.7.2

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

${(\frac{\sqrt{2}}{4 \cdot 2^{\frac{1}{2} \cdot 2}})}^{- \frac{2}{5}}$

Step 3.7.3

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

${(\frac{\sqrt{2}}{4 \cdot 2^{\frac{2}{2}}})}^{- \frac{2}{5}}$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

${(\frac{\sqrt{2}}{4 \cdot 2^{\frac{2}{2}}})}^{- \frac{2}{5}}$

Step 3.7.4.2

Rewrite the expression.

${(\frac{\sqrt{2}}{4 \cdot 2^{1}})}^{- \frac{2}{5}}$

Step 3.7.5

Evaluate the exponent.

${(\frac{\sqrt{2}}{4 \cdot 2})}^{- \frac{2}{5}}$

Step 4

Multiply $4$ by $2$ .

${(\frac{\sqrt{2}}{8})}^{- \frac{2}{5}}$

Step 5

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{8}{\sqrt{2}})}^{\frac{2}{5}}$

Step 6

Multiply $\frac{8}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{8}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{\frac{2}{5}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{8}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{8 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{\frac{2}{5}}$

Step 7.2

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

${(\frac{8 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{\frac{2}{5}}$

Step 7.3

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

${(\frac{8 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{\frac{2}{5}}$

Step 7.4

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

${(\frac{8 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{\frac{2}{5}}$

Step 7.5

Add $1$ and $1$ .

${(\frac{8 \sqrt{2}}{{\sqrt{2}}^{2}})}^{\frac{2}{5}}$

Step 7.6

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

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Step 7.6.1

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

${(\frac{8 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{\frac{2}{5}}$

Step 7.6.2

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

${(\frac{8 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{\frac{2}{5}}$

Step 7.6.3

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

${(\frac{8 \sqrt{2}}{2^{\frac{2}{2}}})}^{\frac{2}{5}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

${(\frac{8 \sqrt{2}}{2^{\frac{2}{2}}})}^{\frac{2}{5}}$

Step 7.6.4.2

Rewrite the expression.

${(\frac{8 \sqrt{2}}{2^{1}})}^{\frac{2}{5}}$

Step 7.6.5

Evaluate the exponent.

${(\frac{8 \sqrt{2}}{2})}^{\frac{2}{5}}$

Step 8

Cancel the common factor of $8$ and $2$ .

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Step 8.1

Factor $2$ out of $8 \sqrt{2}$ .

${(\frac{2 (4 \sqrt{2})}{2})}^{\frac{2}{5}}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $2$ .

${(\frac{2 (4 \sqrt{2})}{2 (1)})}^{\frac{2}{5}}$

Step 8.2.2

Cancel the common factor.

${(\frac{2 (4 \sqrt{2})}{2 \cdot 1})}^{\frac{2}{5}}$

Step 8.2.3

Rewrite the expression.

${(\frac{4 \sqrt{2}}{1})}^{\frac{2}{5}}$

Step 8.2.4

Divide $4 \sqrt{2}$ by $1$ .

${(4 \sqrt{2})}^{\frac{2}{5}}$

Step 9

Apply the product rule to $4 \sqrt{2}$ .

$4^{\frac{2}{5}} {\sqrt{2}}^{\frac{2}{5}}$

Step 10

Rewrite ${\sqrt{2}}^{\frac{2}{5}}$ as $\sqrt[5]{2}$ .

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Step 10.1

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

$4^{\frac{2}{5}} {(2^{\frac{1}{2}})}^{\frac{2}{5}}$

Step 10.2

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

$4^{\frac{2}{5}} \cdot 2^{\frac{1}{2} \cdot \frac{2}{5}}$

Step 10.3

Multiply $\frac{1}{2}$ by $\frac{2}{5}$ .

$4^{\frac{2}{5}} \cdot 2^{\frac{2}{2 \cdot 5}}$

Step 10.4

Multiply $2$ by $5$ .

$4^{\frac{2}{5}} \cdot 2^{\frac{2}{10}}$

Step 10.5

Cancel the common factor of $2$ and $10$ .

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Step 10.5.1

Factor $2$ out of $2$ .

$4^{\frac{2}{5}} \cdot 2^{\frac{2 (1)}{10}}$

Step 10.5.2

Cancel the common factors.

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Step 10.5.2.1

Factor $2$ out of $10$ .

$4^{\frac{2}{5}} \cdot 2^{\frac{2 \cdot 1}{2 \cdot 5}}$

Step 10.5.2.2

Cancel the common factor.

$4^{\frac{2}{5}} \cdot 2^{\frac{2 \cdot 1}{2 \cdot 5}}$

Step 10.5.2.3

Rewrite the expression.

$4^{\frac{2}{5}} \cdot 2^{\frac{1}{5}}$

Step 10.6

Rewrite $2^{\frac{1}{5}}$ as $\sqrt[5]{2}$ .

$4^{\frac{2}{5}} \sqrt[5]{2}$

Step 11

Multiply $4^{\frac{2}{5}} \sqrt[5]{2}$ .

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Step 11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2}$ as $2^{\frac{1}{5}}$ .

$4^{\frac{2}{5}} \cdot 2^{\frac{1}{5}}$

Step 11.2

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

${(2^{2})}^{\frac{2}{5}} \cdot 2^{\frac{1}{5}}$

Step 11.3

Multiply the exponents in ${(2^{2})}^{\frac{2}{5}}$ .

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Step 11.3.1

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

$2^{2 (\frac{2}{5})} \cdot 2^{\frac{1}{5}}$

Step 11.3.2

Multiply $2 (\frac{2}{5})$ .

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Step 11.3.2.1

Combine $2$ and $\frac{2}{5}$ .

$2^{\frac{2 \cdot 2}{5}} \cdot 2^{\frac{1}{5}}$

Step 11.3.2.2

Multiply $2$ by $2$ .

$2^{\frac{4}{5}} \cdot 2^{\frac{1}{5}}$

Step 11.4

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

$2^{\frac{4}{5} + \frac{1}{5}}$

Step 11.5

Combine the numerators over the common denominator.

$2^{\frac{4 + 1}{5}}$

Step 11.6

Add $4$ and $1$ .

$2^{\frac{5}{5}}$

Step 11.7

Cancel the common factor of $5$ .

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Step 11.7.1

Cancel the common factor.

$2^{\frac{5}{5}}$

Step 11.7.2

Rewrite the expression.

$2^{1}$

Step 12

Evaluate the exponent.

$2$