Simplify the Radical Expression ( cube root of a^10b^16)/( cube root of a^5b^4)

$\frac{\sqrt[3]{a^{10} b^{16}}}{\sqrt[3]{a^{5} b^{4}}}$

Step 1

Combine $\sqrt[3]{a^{10} b^{16}}$ and $\sqrt[3]{a^{5} b^{4}}$ into a single radical.

$\sqrt[3]{\frac{a^{10} b^{16}}{a^{5} b^{4}}}$

Step 2

Reduce the expression $\frac{a^{10} b^{16}}{a^{5} b^{4}}$ by cancelling the common factors.

Factor $a^{5}$ out of $a^{10} b^{16}$ .

$\sqrt[3]{\frac{a^{5} (a^{5} b^{16})}{a^{5} b^{4}}}$

Factor $a^{5}$ out of $a^{5} b^{4}$ .

$\sqrt[3]{\frac{a^{5} (a^{5} b^{16})}{a^{5} (b^{4})}}$

Cancel the common factor.

$\sqrt[3]{\frac{a^{5} (a^{5} b^{16})}{a^{5} b^{4}}}$

Rewrite the expression.

$\sqrt[3]{\frac{a^{5} b^{16}}{b^{4}}}$

Step 3

Cancel the common factor of $b^{16}$ and $b^{4}$ .

Factor $b^{4}$ out of $a^{5} b^{16}$ .

$\sqrt[3]{\frac{b^{4} (a^{5} b^{12})}{b^{4}}}$

Cancel the common factors.

Multiply by $1$ .

$\sqrt[3]{\frac{b^{4} (a^{5} b^{12})}{b^{4} \cdot 1}}$

Cancel the common factor.

$\sqrt[3]{\frac{b^{4} (a^{5} b^{12})}{b^{4} \cdot 1}}$

Rewrite the expression.

$\sqrt[3]{\frac{a^{5} b^{12}}{1}}$

Divide $a^{5} b^{12}$ by $1$ .

$\sqrt[3]{a^{5} b^{12}}$

Step 4

Rewrite $a^{5} b^{12}$ as ${(a b^{4})}^{3} a^{2}$ .

Factor out $a^{3}$ .

$\sqrt[3]{a^{3} a^{2} b^{12}}$

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

$\sqrt[3]{a^{3} a^{2} {(b^{4})}^{3}}$

Move $a^{2}$ .

$\sqrt[3]{a^{3} {(b^{4})}^{3} a^{2}}$

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

$\sqrt[3]{{(a b^{4})}^{3} a^{2}}$

Step 5

Pull terms out from under the radical.

$a b^{4} \sqrt[3]{a^{2}}$