Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

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Step 1.1.1

Factor out $x^{4}$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

Raise $\sqrt[4]{x^{3}}$ to the power of $1$ .

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

Step 3.1.3

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

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

Step 3.1.4

Add $1$ and $3$ .

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

Step 3.1.5

Rewrite ${\sqrt[4]{x^{3}}}^{4}$ as $x^{3}$ .

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Step 3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

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

Step 3.1.5.2

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

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

Step 3.1.5.3

Combine $\frac{3}{4}$ and $4$ .

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

Step 3.1.5.4

Multiply $3$ by $4$ .

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

Step 3.1.5.5

Cancel the common factor of $12$ and $4$ .

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Step 3.1.5.5.1

Factor $4$ out of $12$ .

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

Step 3.1.5.5.2

Cancel the common factors.

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Step 3.1.5.5.2.1

Factor $4$ out of $4$ .

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

Step 3.1.5.5.2.2

Cancel the common factor.

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

Step 3.1.5.5.2.3

Rewrite the expression.

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

Step 3.1.5.5.2.4

Divide $3$ by $1$ .

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

Step 3.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[4]{x^{3}}}^{3}$ as $\sqrt[4]{{(x^{3})}^{3}}$ .

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

Step 4.2

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 4.2.1

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

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

Step 4.2.2

Multiply $3$ by $3$ .

$\frac{\sqrt{x} \sqrt[4]{x^{9}}}{x}$

Step 4.3

Rewrite $x^{9}$ as ${(x^{2})}^{4} x$ .

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Step 4.3.1

Factor out $x^{8}$ .

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

Step 4.3.2

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

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

Step 4.4

Pull terms out from under the radical.

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

Step 4.5

Combine exponents.

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Step 4.5.1

Rewrite the expression using the least common index of $4$ .

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Step 4.5.1.1

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

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

Step 4.5.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{2}{4}}$ .

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

Step 4.5.1.3

Rewrite $x^{\frac{2}{4}}$ as $\sqrt[4]{x^{2}}$ .

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

Step 4.5.2

Combine using the product rule for radicals.

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

Step 4.5.3

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

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Step 4.5.3.1

Multiply $x^{2}$ by $x$ .

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Step 4.5.3.1.1

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

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

Step 4.5.3.1.2

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

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

Step 4.5.3.2

Add $2$ and $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

Cancel the common factor.

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

Step 5.2.4

Rewrite the expression.

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

Step 5.2.5

Divide $x \sqrt[4]{x^{3}}$ by $1$ .

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