Simplify ( cube root of 40x)÷( cube root of 5x^2)

$\sqrt[3]{40 x} \div \sqrt[3]{5 x^{2}}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt[3]{40 x}}{\sqrt[3]{5 x^{2}}}$

Step 2

Combine $\sqrt[3]{40 x}$ and $\sqrt[3]{5 x^{2}}$ into a single radical.

$\sqrt[3]{\frac{40 x}{5 x^{2}}}$

Step 3

Reduce the expression $\frac{40 x}{5 x^{2}}$ by cancelling the common factors.

Factor $5$ out of $40 x$ .

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

Factor $5$ out of $5 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

Factor $x$ out of $8 x$ .

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

Cancel the common factors.

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

$\sqrt[3]{\frac{x \cdot 8}{x \cdot x}}$

Cancel the common factor.

$\sqrt[3]{\frac{x \cdot 8}{x \cdot x}}$

Rewrite the expression.

$\sqrt[3]{\frac{8}{x}}$

Step 5

Rewrite $\sqrt[3]{\frac{8}{x}}$ as $\frac{\sqrt[3]{8}}{\sqrt[3]{x}}$ .

$\frac{\sqrt[3]{8}}{\sqrt[3]{x}}$

Step 6

Simplify the numerator.

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

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

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

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

Step 7

Multiply $\frac{2}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

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

Step 8

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

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

Raise $\sqrt[3]{x}$ to the power of $1$ .

$\frac{2 {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1 + 2}}$

Add $1$ and $2$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 {\sqrt[3]{x}}^{2}}{x^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 {\sqrt[3]{x}}^{2}}{x^{1}}$

Simplify.

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

Step 9

Rewrite ${\sqrt[3]{x}}^{2}$ as ${(x^{2})}^{\frac{1}{3}}$ .

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

Simplify ( cube root of 40x)÷( cube root of 5x^2)

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