Algebra Examples

$\sqrt{x^{\frac{3}{4}} x^{- 1}}$

Step 1

Multiply $x^{\frac{3}{4}}$ by $x^{- 1}$ by adding the exponents.

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Step 1.1

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

$\sqrt{x^{\frac{3}{4} - 1}}$

Step 1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{x^{\frac{3}{4} - 1 \cdot \frac{4}{4}}}$

Step 1.3

Combine $- 1$ and $\frac{4}{4}$ .

$\sqrt{x^{\frac{3}{4} + \frac{- 1 \cdot 4}{4}}}$

Step 1.4

Combine the numerators over the common denominator.

$\sqrt{x^{\frac{3 - 1 \cdot 4}{4}}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 1$ by $4$ .

$\sqrt{x^{\frac{3 - 4}{4}}}$

Step 1.5.2

Subtract $4$ from $3$ .

$\sqrt{x^{\frac{- 1}{4}}}$

Step 1.6

Move the negative in front of the fraction.

$\sqrt{x^{- \frac{1}{4}}}$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt{\frac{1}{x^{\frac{1}{4}}}}$

Step 3

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

$\frac{\sqrt{1}}{\sqrt{x^{\frac{1}{4}}}}$

Step 4

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{x^{\frac{1}{4}}}}$

Step 5

Multiply $\frac{1}{\sqrt{x^{\frac{1}{4}}}}$ by $\frac{\sqrt{x^{\frac{1}{4}}}}{\sqrt{x^{\frac{1}{4}}}}$ .

$\frac{1}{\sqrt{x^{\frac{1}{4}}}} \cdot \frac{\sqrt{x^{\frac{1}{4}}}}{\sqrt{x^{\frac{1}{4}}}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{1}{\sqrt{x^{\frac{1}{4}}}}$ by $\frac{\sqrt{x^{\frac{1}{4}}}}{\sqrt{x^{\frac{1}{4}}}}$ .

$\frac{\sqrt{x^{\frac{1}{4}}}}{\sqrt{x^{\frac{1}{4}}} \sqrt{x^{\frac{1}{4}}}}$

Step 6.2

Raise $\sqrt{x^{\frac{1}{4}}}$ to the power of $1$ .

$\frac{\sqrt{x^{\frac{1}{4}}}}{{\sqrt{x^{\frac{1}{4}}}}^{1} \sqrt{x^{\frac{1}{4}}}}$

Step 6.3

Raise $\sqrt{x^{\frac{1}{4}}}$ to the power of $1$ .

$\frac{\sqrt{x^{\frac{1}{4}}}}{{\sqrt{x^{\frac{1}{4}}}}^{1} {\sqrt{x^{\frac{1}{4}}}}^{1}}$

Step 6.4

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

$\frac{\sqrt{x^{\frac{1}{4}}}}{{\sqrt{x^{\frac{1}{4}}}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

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

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{\frac{1}{4}}}$ as $x^{\frac{\frac{1}{4}}{2}}$ .

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

Step 6.6.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.6.3

Multiply $\frac{1}{4} \cdot \frac{1}{2}$ .

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Step 6.6.3.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

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

Step 6.6.3.2

Multiply $4$ by $2$ .

$\frac{\sqrt{x^{\frac{1}{4}}}}{{(x^{\frac{1}{8}})}^{2}}$

Step 6.6.4

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

$\frac{\sqrt{x^{\frac{1}{4}}}}{x^{\frac{1}{8} \cdot 2}}$

Step 6.6.5

Combine $\frac{1}{8}$ and $2$ .

$\frac{\sqrt{x^{\frac{1}{4}}}}{x^{\frac{2}{8}}}$

Step 6.6.6

Cancel the common factor of $2$ and $8$ .

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Step 6.6.6.1

Factor $2$ out of $2$ .

$\frac{\sqrt{x^{\frac{1}{4}}}}{x^{\frac{2 (1)}{8}}}$

Step 6.6.6.2

Cancel the common factors.

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Step 6.6.6.2.1

Factor $2$ out of $8$ .

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

Step 6.6.6.2.2

Cancel the common factor.

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

Step 6.6.6.2.3

Rewrite the expression.

$\frac{\sqrt{x^{\frac{1}{4}}}}{x^{\frac{1}{4}}}$

Step 6.6.7

Rewrite $x^{\frac{1}{4}}$ as $\sqrt[4]{x}$ .

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Add $1$ and $3$ .

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

Step 8.5

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

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Step 8.5.1

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

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

Step 8.5.2

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

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

Step 8.5.3

Combine $\frac{1}{4}$ and $4$ .

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

Step 8.5.4

Cancel the common factor of $4$ .

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Step 8.5.4.1

Cancel the common factor.

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

Step 8.5.4.2

Rewrite the expression.

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

Step 8.5.5

Simplify.

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

Step 9

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

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