Simplify ( fifth root of x^4)/( sixth root of x^4)

$\frac{\sqrt[5]{x^{4}}}{\sqrt[6]{x^{4}}}$

Step 1

Simplify the denominator.

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{\sqrt[5]{x^{4}}}{\sqrt[6]{{(x^{2})}^{2}}}$

Rewrite $\sqrt[6]{{(x^{2})}^{2}}$ as $\sqrt[3]{\sqrt{{(x^{2})}^{2}}}$ .

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

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

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{x^{2}}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{x^{2}}}^{3}$ as $x^{2}$ .

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

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

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

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

Combine $\frac{2}{3}$ and $3$ .

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

Multiply $2$ by $3$ .

$\frac{\sqrt[5]{x^{4}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{6}{3}}}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

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

Step 4

Simplify the numerator.

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

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

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

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

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

Multiply $2$ by $2$ .

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

Factor out $x^{3}$ .

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

Pull terms out from under the radical.

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

Combine exponents.

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{4}}$ as $x^{\frac{4}{5}}$ .

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

Rewrite $x^{\frac{4}{5}}$ as $x^{\frac{12}{15}}$ .

$\frac{x (x^{\frac{12}{15}} \sqrt[3]{x})}{x^{2}}$

Rewrite $x^{\frac{12}{15}}$ as $\sqrt[15]{x^{12}}$ .

$\frac{x (\sqrt[15]{x^{12}} \sqrt[3]{x})}{x^{2}}$

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

$\frac{x (\sqrt[15]{x^{12}} x^{\frac{1}{3}})}{x^{2}}$

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{5}{15}}$ .

$\frac{x (\sqrt[15]{x^{12}} x^{\frac{5}{15}})}{x^{2}}$

Rewrite $x^{\frac{5}{15}}$ as $\sqrt[15]{x^{5}}$ .

$\frac{x (\sqrt[15]{x^{12}} \sqrt[15]{x^{5}})}{x^{2}}$

Combine using the product rule for radicals.

$\frac{x \sqrt[15]{x^{12} x^{5}}}{x^{2}}$

Multiply $x^{12}$ by $x^{5}$ by adding the exponents.

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

$\frac{x \sqrt[15]{x^{12 + 5}}}{x^{2}}$

Add $12$ and $5$ .

$\frac{x \sqrt[15]{x^{17}}}{x^{2}}$

Factor out $x^{15}$ .

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

Pull terms out from under the radical.

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

Step 5

Simplify the numerator.

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

$\frac{x^{1} x \sqrt[15]{x^{2}}}{x^{2}}$

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

$\frac{x^{1} x^{1} \sqrt[15]{x^{2}}}{x^{2}}$

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

$\frac{x^{1 + 1} \sqrt[15]{x^{2}}}{x^{2}}$

Add $1$ and $1$ .

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

Step 6

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

Cancel the common factor.

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

Divide $\sqrt[15]{x^{2}}$ by $1$ .

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

Simplify ( fifth root of x^4)/( sixth root of x^4)

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