Simplify the Radical Expression ( cube root of x^5)/( fifth root of x^2)

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

Step 1

Simplify the numerator.

Factor out $x^{3}$ .

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

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{x \sqrt[3]{x^{2}}}{\sqrt[5]{x^{2}}}$ by $\frac{{\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{4}}$ .

$\frac{x \sqrt[3]{x^{2}}}{\sqrt[5]{x^{2}}} \cdot \frac{{\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{4}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{x \sqrt[3]{x^{2}}}{\sqrt[5]{x^{2}}}$ by $\frac{{\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{4}}$ .

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{\sqrt[5]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}$

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

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{1} {\sqrt[5]{x^{2}}}^{4}}$

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

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{{\sqrt[5]{x^{2}}}^{5}}$

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 $x^{\frac{2}{5}}$ .

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{{(x^{\frac{2}{5}})}^{5}}$

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

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{2}{5} \cdot 5}}$

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

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{2 \cdot 5}{5}}}$

Multiply $2$ by $5$ .

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{10}{5}}}$

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

Factor $5$ out of $10$ .

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{5 \cdot 2}{5}}}$

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{5 \cdot 2}{5 (1)}}}$

Cancel the common factor.

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{5 \cdot 2}{5 \cdot 1}}}$

Rewrite the expression.

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{x^{\frac{2}{1}}}$

Divide $2$ by $1$ .

$\frac{x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{x^{2}}$

Step 4

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

Factor $x$ out of $x \sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}$ .

$\frac{x (\sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4})}{x^{2}}$

Cancel the common factors.

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

$\frac{x (\sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4})}{x \cdot x}$

Cancel the common factor.

$\frac{x (\sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4})}{x \cdot x}$

Rewrite the expression.

$\frac{\sqrt[3]{x^{2}} {\sqrt[5]{x^{2}}}^{4}}{x}$

Step 5

Simplify the numerator.

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

$\frac{\sqrt[3]{x^{2}} \sqrt[5]{{(x^{2})}^{4}}}{x}$

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

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

$\frac{\sqrt[3]{x^{2}} \sqrt[5]{x^{2 \cdot 4}}}{x}$

Multiply $2$ by $4$ .

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

Factor out $x^{5}$ .

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

Pull terms out from under the radical.

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

Combine exponents.

Rewrite the expression using the least common index of $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

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

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{10}{15}}$ .

$\frac{x (x^{\frac{10}{15}} \sqrt[5]{x^{3}})}{x}$

Rewrite $x^{\frac{10}{15}}$ as $\sqrt[15]{x^{10}}$ .

$\frac{x (\sqrt[15]{x^{10}} \sqrt[5]{x^{3}})}{x}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$ .

$\frac{x (\sqrt[15]{x^{10}} x^{\frac{3}{5}})}{x}$

Rewrite $x^{\frac{3}{5}}$ as $x^{\frac{9}{15}}$ .

$\frac{x (\sqrt[15]{x^{10}} x^{\frac{9}{15}})}{x}$

Rewrite $x^{\frac{9}{15}}$ as $\sqrt[15]{x^{9}}$ .

$\frac{x (\sqrt[15]{x^{10}} \sqrt[15]{x^{9}})}{x}$

Combine using the product rule for radicals.

$\frac{x \sqrt[15]{x^{10} x^{9}}}{x}$

Multiply $x^{10}$ by $x^{9}$ by adding the exponents.

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

$\frac{x \sqrt[15]{x^{10 + 9}}}{x}$

Add $10$ and $9$ .

$\frac{x \sqrt[15]{x^{19}}}{x}$

Factor out $x^{15}$ .

$\frac{x \sqrt[15]{x^{15} x^{4}}}{x}$

Pull terms out from under the radical.

$\frac{x (x \sqrt[15]{x^{4}})}{x}$

Step 6

Simplify the numerator.

Raise $x$ to the power of $1$ .

$\frac{x^{1} x \sqrt[15]{x^{4}}}{x}$

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{1} \sqrt[15]{x^{4}}}{x}$

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

$\frac{x^{1 + 1} \sqrt[15]{x^{4}}}{x}$

Add $1$ and $1$ .

$\frac{x^{2} \sqrt[15]{x^{4}}}{x}$

Step 7

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

Factor $x$ out of $x^{2} \sqrt[15]{x^{4}}$ .

$\frac{x (x \sqrt[15]{x^{4}})}{x}$

Cancel the common factors.

Raise $x$ to the power of $1$ .

$\frac{x (x \sqrt[15]{x^{4}})}{x^{1}}$

Factor $x$ out of $x^{1}$ .

$\frac{x (x \sqrt[15]{x^{4}})}{x \cdot 1}$

Cancel the common factor.

$\frac{x (x \sqrt[15]{x^{4}})}{x \cdot 1}$

Rewrite the expression.

$\frac{x \sqrt[15]{x^{4}}}{1}$

Divide $x \sqrt[15]{x^{4}}$ by $1$ .

$x \sqrt[15]{x^{4}}$

Simplify the Radical Expression ( cube root of x^5)/( fifth root of x^2)

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