Calculus Examples

$- 32 {(\sqrt{7})}^{- 5}$

Step 1

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

$- 32 \frac{1}{{\sqrt{7}}^{5}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite ${\sqrt{7}}^{5}$ as $\sqrt{7^{5}}$ .

$- 32 \frac{1}{\sqrt{7^{5}}}$

Step 2.2

Raise $7$ to the power of $5$ .

$- 32 \frac{1}{\sqrt{16807}}$

Step 2.3

Rewrite $16807$ as $49^{2} \cdot 7$ .

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Step 2.3.1

Factor $2401$ out of $16807$ .

$- 32 \frac{1}{\sqrt{2401 (7)}}$

Step 2.3.2

Rewrite $2401$ as $49^{2}$ .

$- 32 \frac{1}{\sqrt{49^{2} \cdot 7}}$

Step 2.4

Pull terms out from under the radical.

$- 32 \frac{1}{49 \sqrt{7}}$

Step 3

Multiply $\frac{1}{49 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$- 32 (\frac{1}{49 \sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{49 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$- 32 \frac{\sqrt{7}}{49 \sqrt{7} \sqrt{7}}$

Step 4.2

Move $\sqrt{7}$ .

$- 32 \frac{\sqrt{7}}{49 (\sqrt{7} \sqrt{7})}$

Step 4.3

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

$- 32 \frac{\sqrt{7}}{49 ({\sqrt{7}}^{1} \sqrt{7})}$

Step 4.4

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

$- 32 \frac{\sqrt{7}}{49 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}$

Step 4.5

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

$- 32 \frac{\sqrt{7}}{49 {\sqrt{7}}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$- 32 \frac{\sqrt{7}}{49 {\sqrt{7}}^{2}}$

Step 4.7

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$- 32 \frac{\sqrt{7}}{49 {(7^{\frac{1}{2}})}^{2}}$

Step 4.7.2

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

$- 32 \frac{\sqrt{7}}{49 \cdot 7^{\frac{1}{2} \cdot 2}}$

Step 4.7.3

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

$- 32 \frac{\sqrt{7}}{49 \cdot 7^{\frac{2}{2}}}$

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

$- 32 \frac{\sqrt{7}}{49 \cdot 7^{\frac{2}{2}}}$

Step 4.7.4.2

Rewrite the expression.

$- 32 \frac{\sqrt{7}}{49 \cdot 7^{1}}$

Step 4.7.5

Evaluate the exponent.

$- 32 \frac{\sqrt{7}}{49 \cdot 7}$

Step 5

Multiply $49$ by $7$ .

$- 32 \frac{\sqrt{7}}{343}$

Step 6

Combine $- 32$ and $\frac{\sqrt{7}}{343}$ .

$\frac{- 32 \sqrt{7}}{343}$

Step 7

Move the negative in front of the fraction.

$- \frac{32 \sqrt{7}}{343}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{32 \sqrt{7}}{343}$

Decimal Form:

$- 0.24683394 \dots$