Simplify ( cube root of 32x^7)÷( cube root of 4x^2)

$\sqrt[3]{32 x^{7}} \div \sqrt[3]{4 x^{2}}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt[3]{32 x^{7}}}{\sqrt[3]{4 x^{2}}}$

Step 2

Combine $\sqrt[3]{32 x^{7}}$ and $\sqrt[3]{4 x^{2}}$ into a single radical.

$\sqrt[3]{\frac{32 x^{7}}{4 x^{2}}}$

Step 3

Reduce the expression $\frac{32 x^{7}}{4 x^{2}}$ by cancelling the common factors.

Factor $4$ out of $32 x^{7}$ .

$\sqrt[3]{\frac{4 (8 x^{7})}{4 x^{2}}}$

Factor $4$ out of $4 x^{2}$ .

$\sqrt[3]{\frac{4 (8 x^{7})}{4 (x^{2})}}$

Cancel the common factor.

$\sqrt[3]{\frac{4 (8 x^{7})}{4 x^{2}}}$

Rewrite the expression.

$\sqrt[3]{\frac{8 x^{7}}{x^{2}}}$

Step 4

Cancel the common factor of $x^{7}$ and $x^{2}$ .

Factor $x^{2}$ out of $8 x^{7}$ .

$\sqrt[3]{\frac{x^{2} (8 x^{5})}{x^{2}}}$

Cancel the common factors.

Multiply by $1$ .

$\sqrt[3]{\frac{x^{2} (8 x^{5})}{x^{2} \cdot 1}}$

Cancel the common factor.

$\sqrt[3]{\frac{x^{2} (8 x^{5})}{x^{2} \cdot 1}}$

Rewrite the expression.

$\sqrt[3]{\frac{8 x^{5}}{1}}$

Divide $8 x^{5}$ by $1$ .

$\sqrt[3]{8 x^{5}}$

Step 5

Rewrite $8 x^{5}$ as ${(2 x)}^{3} x^{2}$ .

Rewrite $8$ as $2^{3}$ .

$\sqrt[3]{2^{3} x^{5}}$

Factor out $x^{3}$ .

$\sqrt[3]{2^{3} (x^{3} x^{2})}$

Rewrite $2^{3} x^{3}$ as ${(2 x)}^{3}$ .

$\sqrt[3]{{(2 x)}^{3} x^{2}}$

Step 6

Pull terms out from under the radical.

$2 x \sqrt[3]{x^{2}}$