Simplify cube root of 21/(-64a^3b^6)

$\sqrt[3]{\frac{21}{- 64 a^{3} b^{6}}}$

Move the negative in front of the fraction.

$\sqrt[3]{- \frac{21}{(64) a^{3} b^{6}}}$

Rewrite $- \frac{21}{64 a^{3} b^{6}}$ as ${(- \frac{1}{4 a b^{2}})}^{3} \cdot 21$ .

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{21}{64 a^{3} b^{6}}}$

Factor the perfect power $1^{3}$ out of $21$ .

$\sqrt[3]{{(- 1)}^{3} \frac{1^{3} \cdot 21}{64 a^{3} b^{6}}}$

Factor the perfect power ${(4 a b^{2})}^{3}$ out of $64 a^{3} b^{6}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{1^{3} \cdot 21}{{(4 a b^{2})}^{3} \cdot 1}}$

Rearrange the fraction $\frac{1^{3} \cdot 21}{{(4 a b^{2})}^{3} \cdot 1}$ .

$\sqrt[3]{{(- 1)}^{3} ({(\frac{1}{4 a b^{2}})}^{3} \cdot 21)}$

Rewrite ${(- 1)}^{3} {(\frac{1}{4 a b^{2}})}^{3}$ as ${(- \frac{1}{4 a b^{2}})}^{3}$ .

$\sqrt[3]{{(- \frac{1}{4 a b^{2}})}^{3} \cdot 21}$

Pull terms out from under the radical.

$- \frac{1}{4 a b^{2}} \sqrt[3]{21}$

Combine $\sqrt[3]{21}$ and $\frac{1}{4 a b^{2}}$ .

$- \frac{\sqrt[3]{21}}{4 a b^{2}}$