Simplify m^2n fifth root of m^5n^12

$m^{2} n \sqrt[5]{m^{5} n^{12}}$

Step 1

Rewrite $m^{5} n^{12}$ as ${(m n^{2})}^{5} n^{2}$ .

Factor out $n^{10}$ .

$m^{2} n \sqrt[5]{m^{5} (n^{10} n^{2})}$

Rewrite $n^{10}$ as ${(n^{2})}^{5}$ .

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

Rewrite $m^{5} {(n^{2})}^{5}$ as ${(m n^{2})}^{5}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

Move $m$ .

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

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

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

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

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

$m^{1 + 2} n (n^{2} \sqrt[5]{n^{2}})$

Add $1$ and $2$ .

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

Step 4

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

Move $n^{2}$ .

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

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

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

$m^{3} (n^{2} n^{1}) \sqrt[5]{n^{2}}$

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

$m^{3} n^{2 + 1} \sqrt[5]{n^{2}}$

Add $2$ and $1$ .

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