Simplify fourth root of 32m^7n^9

$\sqrt[4]{32 m^{7} n^{9}}$

Step 1

Rewrite $32 m^{7} n^{9}$ as ${(2 m n^{2})}^{4} \cdot (2 m^{3} n)$ .

Factor $16$ out of $32$ .

$\sqrt[4]{16 (2) m^{7} n^{9}}$

Rewrite $16$ as $2^{4}$ .

$\sqrt[4]{2^{4} \cdot 2 m^{7} n^{9}}$

Factor out $m^{4}$ .

$\sqrt[4]{2^{4} \cdot 2 (m^{4} m^{3}) n^{9}}$

Factor out $n^{8}$ .

$\sqrt[4]{2^{4} \cdot 2 (m^{4} m^{3}) (n^{8} n)}$

Rewrite $n^{8}$ as ${(n^{2})}^{4}$ .

$\sqrt[4]{2^{4} \cdot 2 (m^{4} m^{3}) ({(n^{2})}^{4} n)}$

Move $m^{3}$ .

$\sqrt[4]{2^{4} \cdot 2 (m^{4}) {(n^{2})}^{4} m^{3} n}$

Move $2$ .

$\sqrt[4]{2^{4} (m^{4}) {(n^{2})}^{4} \cdot 2 m^{3} n}$

Rewrite $(2^{4} (m^{4})) {(n^{2})}^{4}$ as ${(2 m n^{2})}^{4}$ .

$\sqrt[4]{{(2 m n^{2})}^{4} \cdot 2 m^{3} n}$

Add parentheses.

$\sqrt[4]{{(2 m n^{2})}^{4} \cdot 2 (m^{3} n)}$

Add parentheses.

$\sqrt[4]{{(2 m n^{2})}^{4} \cdot (2 m^{3} n)}$

Step 2

Pull terms out from under the radical.

$2 m n^{2} \sqrt[4]{2 m^{3} n}$