Simplify sixth root of 729a^9b^24

$\sqrt[6]{729 a^{9} b^{24}}$

Step 1

Rewrite $729 a^{9} b^{24}$ as ${(3 a b^{4})}^{6} a^{3}$ .

Rewrite $729$ as $3^{6}$ .

$\sqrt[6]{3^{6} a^{9} b^{24}}$

Factor out $a^{6}$ .

$\sqrt[6]{3^{6} (a^{6} a^{3}) b^{24}}$

Rewrite $b^{24}$ as ${(b^{4})}^{6}$ .

$\sqrt[6]{3^{6} (a^{6} a^{3}) {(b^{4})}^{6}}$

Move $a^{3}$ .

$\sqrt[6]{3^{6} (a^{6}) {(b^{4})}^{6} a^{3}}$

Rewrite $3^{6} (a^{6}) {(b^{4})}^{6}$ as ${(3 a b^{4})}^{6}$ .

$\sqrt[6]{{(3 a b^{4})}^{6} a^{3}}$

Step 2

Pull terms out from under the radical.

$3 a b^{4} \sqrt[6]{a^{3}}$

Step 3

Rewrite $\sqrt[6]{a^{3}}$ as $\sqrt{\sqrt[3]{a^{3}}}$ .

$3 a b^{4} \sqrt{\sqrt[3]{a^{3}}}$

Step 4

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

$3 a b^{4} \sqrt{a}$