Simplify cube root of 32x^12y^23

$\sqrt[3]{32 x^{12} y^{23}}$

Step 1

Rewrite $32 x^{12} y^{23}$ as ${(2 x^{4} y^{7})}^{3} \cdot (4 y^{2})$ .

Factor $8$ out of $32$ .

$\sqrt[3]{8 (4) x^{12} y^{23}}$

Rewrite $8$ as $2^{3}$ .

$\sqrt[3]{2^{3} \cdot 4 x^{12} y^{23}}$

Rewrite $x^{12}$ as ${(x^{4})}^{3}$ .

$\sqrt[3]{2^{3} \cdot 4 {(x^{4})}^{3} y^{23}}$

Factor out $y^{21}$ .

$\sqrt[3]{2^{3} \cdot 4 {(x^{4})}^{3} (y^{21} y^{2})}$

Rewrite $y^{21}$ as ${(y^{7})}^{3}$ .

$\sqrt[3]{2^{3} \cdot 4 {(x^{4})}^{3} ({(y^{7})}^{3} y^{2})}$

Move $4$ .

$\sqrt[3]{2^{3} {(x^{4})}^{3} {(y^{7})}^{3} \cdot 4 y^{2}}$

Rewrite $2^{3} {(x^{4})}^{3} {(y^{7})}^{3}$ as ${(2 x^{4} y^{7})}^{3}$ .

$\sqrt[3]{{(2 x^{4} y^{7})}^{3} \cdot 4 y^{2}}$

Add parentheses.

$\sqrt[3]{{(2 x^{4} y^{7})}^{3} \cdot (4 y^{2})}$

Step 2

Pull terms out from under the radical.

$2 x^{4} y^{7} \sqrt[3]{4 y^{2}}$