Simplify cube root of -27a^11b^10

$\sqrt[3]{- 27 a^{11} b^{10}}$

Step 1

Rewrite $- 27 a^{11} b^{10}$ as ${(- 3 a^{3} b^{3})}^{3} (a^{2} b)$ .

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

$\sqrt[3]{{(- 3)}^{3} a^{11} b^{10}}$

Factor out $a^{9}$ .

$\sqrt[3]{{(- 3)}^{3} (a^{9} a^{2}) b^{10}}$

Rewrite $a^{9}$ as ${(a^{3})}^{3}$ .

$\sqrt[3]{{(- 3)}^{3} ({(a^{3})}^{3} a^{2}) b^{10}}$

Factor out $b^{9}$ .

$\sqrt[3]{{(- 3)}^{3} ({(a^{3})}^{3} a^{2}) (b^{9} b)}$

Rewrite $b^{9}$ as ${(b^{3})}^{3}$ .

$\sqrt[3]{{(- 3)}^{3} ({(a^{3})}^{3} a^{2}) ({(b^{3})}^{3} b)}$

Move $a^{2}$ .

$\sqrt[3]{{(- 3)}^{3} ({(a^{3})}^{3}) {(b^{3})}^{3} a^{2} b}$

Rewrite ${(- 3)}^{3} ({(a^{3})}^{3}) {(b^{3})}^{3}$ as ${(- 3 a^{3} b^{3})}^{3}$ .

$\sqrt[3]{{(- 3 a^{3} b^{3})}^{3} a^{2} b}$

Add parentheses.

$\sqrt[3]{{(- 3 a^{3} b^{3})}^{3} (a^{2} b)}$

Step 2

Pull terms out from under the radical.

$- 3 a^{3} b^{3} \sqrt[3]{a^{2} b}$