Simplify ( fourth root of x^5)/( sixth root of x^5)

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

Step 1

Simplify the numerator.

Factor out $x^{4}$ .

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

Combine and simplify the denominator.

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

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

Raise $\sqrt[6]{x^{5}}$ to the power of $1$ .

$\frac{x \sqrt[4]{x} {\sqrt[6]{x^{5}}}^{5}}{{\sqrt[6]{x^{5}}}^{1} {\sqrt[6]{x^{5}}}^{5}}$

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

$\frac{x \sqrt[4]{x} {\sqrt[6]{x^{5}}}^{5}}{{\sqrt[6]{x^{5}}}^{1 + 5}}$

Add $1$ and $5$ .

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

Rewrite ${\sqrt[6]{x^{5}}}^{6}$ as $x^{5}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x^{5}}$ as $x^{\frac{5}{6}}$ .

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

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

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

Combine $\frac{5}{6}$ and $6$ .

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

Multiply $5$ by $6$ .

$\frac{x \sqrt[4]{x} {\sqrt[6]{x^{5}}}^{5}}{x^{\frac{30}{6}}}$

Cancel the common factor of $30$ and $6$ .

Factor $6$ out of $30$ .

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

Cancel the common factors.

Factor $6$ out of $6$ .

$\frac{x \sqrt[4]{x} {\sqrt[6]{x^{5}}}^{5}}{x^{\frac{6 \cdot 5}{6 (1)}}}$

Cancel the common factor.

$\frac{x \sqrt[4]{x} {\sqrt[6]{x^{5}}}^{5}}{x^{\frac{6 \cdot 5}{6 \cdot 1}}}$

Rewrite the expression.

$\frac{x \sqrt[4]{x} {\sqrt[6]{x^{5}}}^{5}}{x^{\frac{5}{1}}}$

Divide $5$ by $1$ .

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

Cancel the common factor of $x$ and $x^{5}$ .

Factor $x$ out of $x \sqrt[4]{x} {\sqrt[6]{x^{5}}}^{5}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Rewrite ${\sqrt[6]{x^{5}}}^{5}$ as $\sqrt[6]{{(x^{5})}^{5}}$ .

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

Multiply the exponents in ${(x^{5})}^{5}$ .

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

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

Multiply $5$ by $5$ .

$\frac{\sqrt[4]{x} \sqrt[6]{x^{25}}}{x^{4}}$

Rewrite $x^{25}$ as ${(x^{4})}^{6} x$ .

Factor out $x^{24}$ .

$\frac{\sqrt[4]{x} \sqrt[6]{x^{24} x}}{x^{4}}$

Rewrite $x^{24}$ as ${(x^{4})}^{6}$ .

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

Pull terms out from under the radical.

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

Combine exponents.

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\frac{x^{4} (x^{\frac{1}{4}} \sqrt[6]{x})}{x^{4}}$

Rewrite $x^{\frac{1}{4}}$ as $x^{\frac{3}{12}}$ .

$\frac{x^{4} (x^{\frac{3}{12}} \sqrt[6]{x})}{x^{4}}$

Rewrite $x^{\frac{3}{12}}$ as $\sqrt[12]{x^{3}}$ .

$\frac{x^{4} (\sqrt[12]{x^{3}} \sqrt[6]{x})}{x^{4}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$\frac{x^{4} (\sqrt[12]{x^{3}} x^{\frac{1}{6}})}{x^{4}}$

Rewrite $x^{\frac{1}{6}}$ as $x^{\frac{2}{12}}$ .

$\frac{x^{4} (\sqrt[12]{x^{3}} x^{\frac{2}{12}})}{x^{4}}$

Rewrite $x^{\frac{2}{12}}$ as $\sqrt[12]{x^{2}}$ .

$\frac{x^{4} (\sqrt[12]{x^{3}} \sqrt[12]{x^{2}})}{x^{4}}$

Combine using the product rule for radicals.

$\frac{x^{4} \sqrt[12]{x^{3} x^{2}}}{x^{4}}$

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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

$\frac{x^{4} \sqrt[12]{x^{3 + 2}}}{x^{4}}$

Add $3$ and $2$ .

$\frac{x^{4} \sqrt[12]{x^{5}}}{x^{4}}$

Step 5

Cancel the common factor of $x^{4}$ .

Cancel the common factor.

$\frac{x^{4} \sqrt[12]{x^{5}}}{x^{4}}$

Divide $\sqrt[12]{x^{5}}$ by $1$ .

$\sqrt[12]{x^{5}}$

Simplify ( fourth root of x^5)/( sixth root of x^5)

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