Precalculus Examples

\sqrt{s \sqrt{s^{7}}}

$\sqrt{s \sqrt{s^{7}}}$

Step 1

Rewrite

s^{7}

$s^{7}$ as

{(s^{3})}^{2} s

${(s^{3})}^{2} s$ .

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Step 1.1

Factor out

s^{6}

$s^{6}$ .

\sqrt{s \sqrt{s^{6} s}}

$\sqrt{s \sqrt{s^{6} s}}$

Step 1.2

Rewrite

s^{6}

$s^{6}$ as

{(s^{3})}^{2}

${(s^{3})}^{2}$ .

\sqrt{s \sqrt{{(s^{3})}^{2} s}}

$\sqrt{s \sqrt{{(s^{3})}^{2} s}}$

\sqrt{s \sqrt{{(s^{3})}^{2} s}}

$\sqrt{s \sqrt{{(s^{3})}^{2} s}}$

Step 2

Pull terms out from under the radical.

\sqrt{s \cdot s^{3} \sqrt{s}}

$\sqrt{s \cdot s^{3} \sqrt{s}}$

Step 3

Multiply

s

$s$ by

s^{3}

$s^{3}$ by adding the exponents.

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Step 3.1

Multiply

s

$s$ by

s^{3}

$s^{3}$ .

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Step 3.1.1

Raise

s

$s$ to the power of

1

$1$ .

\sqrt{s^{1} s^{3} \sqrt{s}}

$\sqrt{s^{1} s^{3} \sqrt{s}}$

Step 3.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\sqrt{s^{1 + 3} \sqrt{s}}

$\sqrt{s^{1 + 3} \sqrt{s}}$

\sqrt{s^{1 + 3} \sqrt{s}}

$\sqrt{s^{1 + 3} \sqrt{s}}$

Step 3.2

Add

1

$1$ and

3

$3$ .

\sqrt{s^{4} \sqrt{s}}

$\sqrt{s^{4} \sqrt{s}}$

\sqrt{s^{4} \sqrt{s}}

$\sqrt{s^{4} \sqrt{s}}$

Step 4

Rewrite

s^{4}

$s^{4}$ as

{(s^{2})}^{2}

${(s^{2})}^{2}$ .

\sqrt{{(s^{2})}^{2} \sqrt{s}}

$\sqrt{{(s^{2})}^{2} \sqrt{s}}$

Step 5

Pull terms out from under the radical.

s^{2} \sqrt{\sqrt{s}}

$s^{2} \sqrt{\sqrt{s}}$

Step 6

Rewrite

\sqrt{\sqrt{s}}

$\sqrt{\sqrt{s}}$ as

\sqrt[4]{s}

$\sqrt[4]{s}$ .

s^{2} \sqrt[4]{s}

$s^{2} \sqrt[4]{s}$

\sqrt[]{s \sqrt[]{s^{7}}}

$\sqrt[]{s \sqrt[]{s^{7}}}$

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[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$