Algebra Examples

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

Step 1

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

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

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

Step 1.2

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

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

Step 1.3

Multiply $\frac{1}{5}$ by $\frac{5}{6}$ .

$x^{\frac{5}{5 \cdot 6}} \cdot \sqrt{x}$

Step 1.4

Multiply $5$ by $6$ .

$x^{\frac{5}{30}} \cdot \sqrt{x}$

Step 1.5

Cancel the common factor of $5$ and $30$ .

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Step 1.5.1

Factor $5$ out of $5$ .

$x^{\frac{5 (1)}{30}} \cdot \sqrt{x}$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $5$ out of $30$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

$x^{\frac{1}{6}} \cdot \sqrt{x}$

Step 1.6

Rewrite $x^{\frac{1}{6}}$ as $\sqrt[6]{x}$ .

$\sqrt[6]{x} \cdot \sqrt{x}$

Step 2

Multiply $\sqrt[6]{x} \sqrt{x}$ .

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Step 2.1

Rewrite the expression using the least common index of $6$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\sqrt[6]{x} x^{\frac{1}{2}}$

Step 2.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

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

Step 2.1.3

Rewrite $x^{\frac{3}{6}}$ as $\sqrt[6]{x^{3}}$ .

$\sqrt[6]{x} \sqrt[6]{x^{3}}$

Step 2.2

Combine using the product rule for radicals.

$\sqrt[6]{x \cdot x^{3}}$

Step 2.3

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

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Step 2.3.1

Multiply $x$ by $x^{3}$ .

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Step 2.3.1.1

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

$\sqrt[6]{x^{1} x^{3}}$

Step 2.3.1.2

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

$\sqrt[6]{x^{1 + 3}}$

Step 2.3.2

Add $1$ and $3$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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