Evaluate ( cube root of x^3)/( fifth root of x^2)

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

Step 1

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

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

Step 2

Multiply $\frac{x}{\sqrt[5]{x^{2}}}$ by $\frac{{\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{4}}$ .

$\frac{x}{\sqrt[5]{x^{2}}} \cdot \frac{{\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{4}}$

Step 3

Simplify terms.

Combine and simplify the denominator.

Multiply $\frac{x}{\sqrt[5]{x^{2}}}$ by $\frac{{\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{4}}$ .

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{\sqrt[5]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}$

Raise $\sqrt[5]{x^{2}}$ to the power of $1$ .

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{1} {\sqrt[5]{x^{2}}}^{4}}$

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

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{5}}$

Rewrite ${\sqrt[5]{x^{2}}}^{5}$ as $x^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{{(x^{\frac{2}{5}})}^{5}}$

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

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{2}{5} \cdot 5}}$

Combine $\frac{2}{5}$ and $5$ .

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{2 \cdot 5}{5}}}$

Multiply $2$ by $5$ .

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{10}{5}}}$

Cancel the common factor of $10$ and $5$ .

Factor $5$ out of $10$ .

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{5 \cdot 2}{5}}}$

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{5 \cdot 2}{5 (1)}}}$

Cancel the common factor.

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{5 \cdot 2}{5 \cdot 1}}}$

Rewrite the expression.

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{2}{1}}}$

Divide $2$ by $1$ .

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{x^{2}}$

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

Factor $x$ out of $x {\sqrt[5]{x^{2}}}^{4}$ .

$\frac{x ({\sqrt[5]{x^{2}}}^{4})}{x^{2}}$

Cancel the common factors.

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

$\frac{x ({\sqrt[5]{x^{2}}}^{4})}{x \cdot x}$

Cancel the common factor.

$\frac{x {\sqrt[5]{x^{2}}}^{4}}{x \cdot x}$

Rewrite the expression.

$\frac{{\sqrt[5]{x^{2}}}^{4}}{x}$

Step 4

Simplify the numerator.

Rewrite ${\sqrt[5]{x^{2}}}^{4}$ as ${({(x^{2})}^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{{(x^{2})}^{4}}}{x}$

Multiply the exponents in ${(x^{2})}^{4}$ .

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

$\frac{\sqrt[5]{x^{2 \cdot 4}}}{x}$

Multiply $2$ by $4$ .

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

Factor out $x^{5}$ .

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

Pull terms out from under the radical.

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

Step 5

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $\sqrt[5]{x^{3}}$ by $1$ .

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

Evaluate ( cube root of x^3)/( fifth root of x^2)

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