Calculus Examples

$\sqrt[5]{\sqrt{3^{- 25}}}$

Step 1

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

$\sqrt[5]{\sqrt{\frac{1}{3^{25}}}}$

Step 2

Raise $3$ to the power of $25$ .

$\sqrt[5]{\sqrt{\frac{1}{847288609443}}}$

Step 3

Rewrite $\sqrt{\frac{1}{847288609443}}$ as $\frac{\sqrt{1}}{\sqrt{847288609443}}$ .

$\sqrt[5]{\frac{\sqrt{1}}{\sqrt{847288609443}}}$

Step 4

Any root of $1$ is $1$ .

$\sqrt[5]{\frac{1}{\sqrt{847288609443}}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $847288609443$ as $531441^{2} \cdot 3$ .

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Step 5.1.1

Factor $282429536481$ out of $847288609443$ .

$\sqrt[5]{\frac{1}{\sqrt{282429536481 (3)}}}$

Step 5.1.2

Rewrite $282429536481$ as $531441^{2}$ .

$\sqrt[5]{\frac{1}{\sqrt{531441^{2} \cdot 3}}}$

Step 5.2

Pull terms out from under the radical.

$\sqrt[5]{\frac{1}{531441 \sqrt{3}}}$

Step 6

Multiply $\frac{1}{531441 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt[5]{\frac{1}{531441 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{531441 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt[5]{\frac{\sqrt{3}}{531441 \sqrt{3} \sqrt{3}}}$

Step 7.2

Move $\sqrt{3}$ .

$\sqrt[5]{\frac{\sqrt{3}}{531441 (\sqrt{3} \sqrt{3})}}$

Step 7.3

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt[5]{\frac{\sqrt{3}}{531441 ({\sqrt{3}}^{1} \sqrt{3})}}$

Step 7.4

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt[5]{\frac{\sqrt{3}}{531441 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}}$

Step 7.5

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

$\sqrt[5]{\frac{\sqrt{3}}{531441 {\sqrt{3}}^{1 + 1}}}$

Step 7.6

Add $1$ and $1$ .

$\sqrt[5]{\frac{\sqrt{3}}{531441 {\sqrt{3}}^{2}}}$

Step 7.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt[5]{\frac{\sqrt{3}}{531441 {(3^{\frac{1}{2}})}^{2}}}$

Step 7.7.2

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

$\sqrt[5]{\frac{\sqrt{3}}{531441 \cdot 3^{\frac{1}{2} \cdot 2}}}$

Step 7.7.3

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

$\sqrt[5]{\frac{\sqrt{3}}{531441 \cdot 3^{\frac{2}{2}}}}$

Step 7.7.4

Cancel the common factor of $2$ .

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Step 7.7.4.1

Cancel the common factor.

$\sqrt[5]{\frac{\sqrt{3}}{531441 \cdot 3^{\frac{2}{2}}}}$

Step 7.7.4.2

Rewrite the expression.

$\sqrt[5]{\frac{\sqrt{3}}{531441 \cdot 3^{1}}}$

Step 7.7.5

Evaluate the exponent.

$\sqrt[5]{\frac{\sqrt{3}}{531441 \cdot 3}}$

Step 8

Multiply $531441$ by $3$ .

$\sqrt[5]{\frac{\sqrt{3}}{1594323}}$

Step 9

Rewrite $\sqrt[5]{\frac{\sqrt{3}}{1594323}}$ as $\frac{\sqrt[5]{\sqrt{3}}}{\sqrt[5]{1594323}}$ .

$\frac{\sqrt[5]{\sqrt{3}}}{\sqrt[5]{1594323}}$

Step 10

Rewrite $\sqrt[5]{\sqrt{3}}$ as $\sqrt[10]{3}$ .

$\frac{\sqrt[10]{3}}{\sqrt[5]{1594323}}$

Step 11

Simplify the denominator.

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Step 11.1

Rewrite $1594323$ as $9^{5} \cdot 27$ .

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Step 11.1.1

Factor $59049$ out of $1594323$ .

$\frac{\sqrt[10]{3}}{\sqrt[5]{59049 (27)}}$

Step 11.1.2

Rewrite $59049$ as $9^{5}$ .

$\frac{\sqrt[10]{3}}{\sqrt[5]{9^{5} \cdot 27}}$

Step 11.2

Pull terms out from under the radical.

$\frac{\sqrt[10]{3}}{9 \sqrt[5]{27}}$

Step 12

Multiply $\frac{\sqrt[10]{3}}{9 \sqrt[5]{27}}$ by $\frac{{\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{4}}$ .

$\frac{\sqrt[10]{3}}{9 \sqrt[5]{27}} \cdot \frac{{\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{4}}$

Step 13

Combine and simplify the denominator.

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Step 13.1

Multiply $\frac{\sqrt[10]{3}}{9 \sqrt[5]{27}}$ by $\frac{{\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{4}}$ .

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 \sqrt[5]{27} {\sqrt[5]{27}}^{4}}$

Step 13.2

Move $\sqrt[5]{27}$ .

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 (\sqrt[5]{27} {\sqrt[5]{27}}^{4})}$

Step 13.3

Raise $\sqrt[5]{27}$ to the power of $1$ .

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 ({\sqrt[5]{27}}^{1} {\sqrt[5]{27}}^{4})}$

Step 13.4

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

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 {\sqrt[5]{27}}^{1 + 4}}$

Step 13.5

Add $1$ and $4$ .

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 {\sqrt[5]{27}}^{5}}$

Step 13.6

Rewrite ${\sqrt[5]{27}}^{5}$ as $27$ .

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Step 13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{27}$ as $27^{\frac{1}{5}}$ .

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 {(27^{\frac{1}{5}})}^{5}}$

Step 13.6.2

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

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 \cdot 27^{\frac{1}{5} \cdot 5}}$

Step 13.6.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 \cdot 27^{\frac{5}{5}}}$

Step 13.6.4

Cancel the common factor of $5$ .

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Step 13.6.4.1

Cancel the common factor.

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 \cdot 27^{\frac{5}{5}}}$

Step 13.6.4.2

Rewrite the expression.

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 \cdot 27^{1}}$

Step 13.6.5

Evaluate the exponent.

$\frac{\sqrt[10]{3} {\sqrt[5]{27}}^{4}}{9 \cdot 27}$

Step 14

Simplify the numerator.

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Step 14.1

Rewrite ${\sqrt[5]{27}}^{4}$ as $\sqrt[5]{27^{4}}$ .

$\frac{\sqrt[10]{3} \sqrt[5]{27^{4}}}{9 \cdot 27}$

Step 14.2

Raise $27$ to the power of $4$ .

$\frac{\sqrt[10]{3} \sqrt[5]{531441}}{9 \cdot 27}$

Step 14.3

Rewrite $531441$ as $9^{5} \cdot 9$ .

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Step 14.3.1

Factor $59049$ out of $531441$ .

$\frac{\sqrt[10]{3} \sqrt[5]{59049 (9)}}{9 \cdot 27}$

Step 14.3.2

Rewrite $59049$ as $9^{5}$ .

$\frac{\sqrt[10]{3} \sqrt[5]{9^{5} \cdot 9}}{9 \cdot 27}$

Step 14.4

Pull terms out from under the radical.

$\frac{\sqrt[10]{3} \cdot 9 \sqrt[5]{9}}{9 \cdot 27}$

Step 14.5

Combine exponents.

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Step 14.5.1

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

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Step 14.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{9}$ as $9^{\frac{1}{5}}$ .

$\frac{9 (\sqrt[10]{3} \cdot 9^{\frac{1}{5}})}{9 \cdot 27}$

Step 14.5.1.2

Rewrite $9^{\frac{1}{5}}$ as $9^{\frac{2}{10}}$ .

$\frac{9 (\sqrt[10]{3} \cdot 9^{\frac{2}{10}})}{9 \cdot 27}$

Step 14.5.1.3

Rewrite $9^{\frac{2}{10}}$ as $\sqrt[10]{9^{2}}$ .

$\frac{9 (\sqrt[10]{3} \sqrt[10]{9^{2}})}{9 \cdot 27}$

Step 14.5.2

Combine using the product rule for radicals.

$\frac{9 \sqrt[10]{3 \cdot 9^{2}}}{9 \cdot 27}$

Step 14.5.3

Rewrite $9$ as $3^{2}$ .

$\frac{9 \sqrt[10]{3 \cdot {(3^{2})}^{2}}}{9 \cdot 27}$

Step 14.5.4

Multiply the exponents in ${(3^{2})}^{2}$ .

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Step 14.5.4.1

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

$\frac{9 \sqrt[10]{3 \cdot 3^{2 \cdot 2}}}{9 \cdot 27}$

Step 14.5.4.2

Multiply $2$ by $2$ .

$\frac{9 \sqrt[10]{3 \cdot 3^{4}}}{9 \cdot 27}$

Step 14.5.5

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

$\frac{9 \sqrt[10]{3^{1 + 4}}}{9 \cdot 27}$

Step 14.5.6

Add $1$ and $4$ .

$\frac{9 \sqrt[10]{3^{5}}}{9 \cdot 27}$

Step 14.6

Raise $3$ to the power of $5$ .

$\frac{9 \sqrt[10]{243}}{9 \cdot 27}$

Step 14.7

Rewrite $243$ as $3^{5}$ .

$\frac{9 \sqrt[10]{3^{5}}}{9 \cdot 27}$

Step 14.8

Rewrite $\sqrt[10]{3^{5}}$ as $\sqrt{\sqrt[5]{3^{5}}}$ .

$\frac{9 \sqrt{\sqrt[5]{3^{5}}}}{9 \cdot 27}$

Step 14.9

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

$\frac{9 \sqrt{3}}{9 \cdot 27}$

Step 15

Reduce the expression by cancelling the common factors.

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Step 15.1

Multiply $9$ by $27$ .

$\frac{9 \sqrt{3}}{243}$

Step 15.2

Cancel the common factor of $9$ and $243$ .

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Step 15.2.1

Factor $9$ out of $9 \sqrt{3}$ .

$\frac{9 (\sqrt{3})}{243}$

Step 15.2.2

Cancel the common factors.

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Step 15.2.2.1

Factor $9$ out of $243$ .

$\frac{9 \sqrt{3}}{9 \cdot 27}$

Step 15.2.2.2

Cancel the common factor.

$\frac{9 \sqrt{3}}{9 \cdot 27}$

Step 15.2.2.3

Rewrite the expression.

$\frac{\sqrt{3}}{27}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{27}$

Decimal Form:

$0.06415002 \dots$