Simplify sixth root of s^7 square root of s^5

$\sqrt[6]{s^{7}} \sqrt{s^{5}}$

Step 1

Factor out $s^{6}$ .

$\sqrt[6]{s^{6} s} \sqrt{s^{5}}$

Step 2

Pull terms out from under the radical.

$s \sqrt[6]{s} \sqrt{s^{5}}$

Step 3

Rewrite $s^{5}$ as ${(s^{2})}^{2} s$ .

Factor out $s^{4}$ .

$s \sqrt[6]{s} \sqrt{s^{4} s}$

Rewrite $s^{4}$ as ${(s^{2})}^{2}$ .

$s \sqrt[6]{s} \sqrt{{(s^{2})}^{2} s}$

Step 4

Pull terms out from under the radical.

$s \sqrt[6]{s} (s^{2} \sqrt{s})$

Step 5

Multiply $s$ by $s^{2}$ by adding the exponents.

Move $s^{2}$ .

$s^{2} s \sqrt[6]{s} \sqrt{s}$

Multiply $s^{2}$ by $s$ .

Raise $s$ to the power of $1$ .

$s^{2} s^{1} \sqrt[6]{s} \sqrt{s}$

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

$s^{2 + 1} \sqrt[6]{s} \sqrt{s}$

Add $2$ and $1$ .

$s^{3} \sqrt[6]{s} \sqrt{s}$

Step 6

Multiply $s^{3} \sqrt[6]{s} \sqrt{s}$ .

Rewrite the expression using the least common index of $6$ .

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

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

Rewrite $s^{\frac{1}{2}}$ as $s^{\frac{3}{6}}$ .

$s^{3} (s^{\frac{3}{6}} \sqrt[6]{s})$

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

$s^{3} (\sqrt[6]{s^{3}} \sqrt[6]{s})$

Combine using the product rule for radicals.

$s^{3} \sqrt[6]{s^{3} s}$

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

Multiply $s^{3}$ by $s$ .

Raise $s$ to the power of $1$ .

$s^{3} \sqrt[6]{s^{3} s^{1}}$

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

$s^{3} \sqrt[6]{s^{3 + 1}}$

Add $3$ and $1$ .

$s^{3} \sqrt[6]{s^{4}}$

Step 7

Rewrite $s^{4}$ as ${(s^{2})}^{2}$ .

$s^{3} \sqrt[6]{{(s^{2})}^{2}}$

Step 8

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

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

Step 9

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

$s^{3} \sqrt[3]{s^{2}}$

Simplify sixth root of s^7 square root of s^5

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