Simplify cube root of (75a^7b^4)/(40a^13c^9)

$\sqrt[3]{\frac{75 a^{7} b^{4}}{40 a^{13} c^{9}}}$

Step 1

Reduce the expression $\frac{75 a^{7} b^{4}}{40 a^{13} c^{9}}$ by cancelling the common factors.

Factor $5$ out of $75 a^{7} b^{4}$ .

$\sqrt[3]{\frac{5 (15 a^{7} b^{4})}{40 a^{13} c^{9}}}$

Factor $5$ out of $40 a^{13} c^{9}$ .

$\sqrt[3]{\frac{5 (15 a^{7} b^{4})}{5 (8 a^{13} c^{9})}}$

Cancel the common factor.

$\sqrt[3]{\frac{5 (15 a^{7} b^{4})}{5 (8 a^{13} c^{9})}}$

Rewrite the expression.

$\sqrt[3]{\frac{15 a^{7} b^{4}}{8 a^{13} c^{9}}}$

Step 2

Cancel the common factor of $a^{7}$ and $a^{13}$ .

Factor $a^{7}$ out of $15 a^{7} b^{4}$ .

$\sqrt[3]{\frac{a^{7} (15 b^{4})}{8 a^{13} c^{9}}}$

Cancel the common factors.

Factor $a^{7}$ out of $8 a^{13} c^{9}$ .

$\sqrt[3]{\frac{a^{7} (15 b^{4})}{a^{7} (8 a^{6} c^{9})}}$

Cancel the common factor.

$\sqrt[3]{\frac{a^{7} (15 b^{4})}{a^{7} (8 a^{6} c^{9})}}$

Rewrite the expression.

$\sqrt[3]{\frac{15 b^{4}}{8 a^{6} c^{9}}}$

Step 3

Rewrite $\frac{15 b^{4}}{8 a^{6} c^{9}}$ as ${(\frac{b}{2 a^{2} c^{3}})}^{3} (15 b)$ .

Factor the perfect power $b^{3}$ out of $15 b^{4}$ .

$\sqrt[3]{\frac{b^{3} \cdot (15 b)}{8 a^{6} c^{9}}}$

Factor the perfect power ${(2 a^{2} c^{3})}^{3}$ out of $8 a^{6} c^{9}$ .

$\sqrt[3]{\frac{b^{3} \cdot (15 b)}{{(2 a^{2} c^{3})}^{3} \cdot 1}}$

Rearrange the fraction $\frac{b^{3} \cdot (15 b)}{{(2 a^{2} c^{3})}^{3} \cdot 1}$ .

$\sqrt[3]{{(\frac{b}{2 a^{2} c^{3}})}^{3} (15 b)}$

Step 4

Pull terms out from under the radical.

$\frac{b}{2 a^{2} c^{3}} \sqrt[3]{15 b}$

Step 5

Combine $\frac{b}{2 a^{2} c^{3}}$ and $\sqrt[3]{15 b}$ .

$\frac{b \sqrt[3]{15 b}}{2 a^{2} c^{3}}$