Trigonometry Examples

$\sqrt[6]{3 m^{5} n^{8}} \sqrt[6]{9 m n}$

Step 1

Rewrite $3 m^{5} n^{8}$ as $n^{6} \cdot (3 m^{5} n^{2})$ .

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Step 1.1

Factor out $n^{6}$ .

$\sqrt[6]{3 m^{5} (n^{6} n^{2})} \sqrt[6]{9 m n}$

Step 1.2

Move $m^{5}$ .

$\sqrt[6]{3 n^{6} m^{5} n^{2}} \sqrt[6]{9 m n}$

Step 1.3

Reorder $3$ and $n^{6}$ .

$\sqrt[6]{n^{6} \cdot 3 m^{5} n^{2}} \sqrt[6]{9 m n}$

Step 1.4

Add parentheses.

$\sqrt[6]{n^{6} \cdot 3 (m^{5} n^{2})} \sqrt[6]{9 m n}$

Step 1.5

Add parentheses.

$\sqrt[6]{n^{6} \cdot (3 m^{5} n^{2})} \sqrt[6]{9 m n}$

Step 2

Pull terms out from under the radical.

$n \sqrt[6]{3 m^{5} n^{2}} \sqrt[6]{9 m n}$

Step 3

Multiply $n \sqrt[6]{3 m^{5} n^{2}} \sqrt[6]{9 m n}$ .

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Step 3.1

Combine using the product rule for radicals.

$n \sqrt[6]{9 m n (3 m^{5} n^{2})}$

Step 3.2

Multiply $3$ by $9$ .

$n \sqrt[6]{27 m n (m^{5} n^{2})}$

Step 3.3

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

$n \sqrt[6]{27 (m^{5} m^{1}) n \cdot n^{2}}$

Step 3.4

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

$n \sqrt[6]{27 m^{5 + 1} n \cdot n^{2}}$

Step 3.5

Add $5$ and $1$ .

$n \sqrt[6]{27 m^{6} n \cdot n^{2}}$

Step 3.6

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

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Step 3.6.1

Move $n^{2}$ .

$n \sqrt[6]{27 m^{6} (n^{2} n)}$

Step 3.6.2

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

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Step 3.6.2.1

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

$n \sqrt[6]{27 m^{6} (n^{2} n^{1})}$

Step 3.6.2.2

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

$n \sqrt[6]{27 m^{6} n^{2 + 1}}$

Step 3.6.3

Add $2$ and $1$ .

$n \sqrt[6]{27 m^{6} n^{3}}$

Step 4

Rewrite $27 m^{6} n^{3}$ as $m^{6} \cdot (27 n^{3})$ .

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Step 4.1

Reorder $27$ and $m^{6}$ .

$n \sqrt[6]{m^{6} \cdot 27 n^{3}}$

Step 4.2

Add parentheses.

$n \sqrt[6]{m^{6} \cdot (27 n^{3})}$

Step 5

Pull terms out from under the radical.

$n (m \sqrt[6]{27 n^{3}})$

Step 6

Rewrite $27 n^{3}$ as ${(3 n)}^{3}$ .

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

Step 7

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

$n (m \sqrt{\sqrt[3]{{(3 n)}^{3}}})$

Step 8

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

$n (m \sqrt{3 n})$

Step 9

Reorder factors in $n (m \sqrt{3 n})$ .

$n m \sqrt{3 n}$