Algebra Examples

$\frac{\sqrt[3]{x^{4}}}{\sqrt[8]{x^{4}}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor out $x^{3}$ .

$\frac{\sqrt[3]{x^{3} x}}{\sqrt[8]{x^{4}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{x \sqrt[3]{x}}{\sqrt[8]{x^{4}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $\sqrt[8]{x^{4}}$ as $\sqrt{\sqrt[4]{x^{4}}}$ .

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

Step 2.2

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

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

Step 3

Multiply $\frac{x \sqrt[3]{x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{x \sqrt[3]{x}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{x \sqrt[3]{x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 4.2

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 4.3

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Simplify.

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

Step 5

Cancel the common factor of $x$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

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

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

Step 6

Multiply $\sqrt[3]{x} \sqrt{x}$ .

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Step 6.1

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

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Step 6.1.1

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

$x^{\frac{1}{3}} \sqrt{x}$

Step 6.1.2

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

$x^{\frac{2}{6}} \sqrt{x}$

Step 6.1.3

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

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

Step 6.1.4

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

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

Step 6.1.5

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

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

Step 6.1.6

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

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

Step 6.2

Combine using the product rule for radicals.

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

Step 6.3

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

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Step 6.3.1

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

$\sqrt[6]{x^{2 + 3}}$

Step 6.3.2

Add $2$ and $3$ .

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