Algebra Examples

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

Step 1

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

${(\frac{\sqrt[3]{x^{2}}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})}^{- 6}$

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

${(\frac{\sqrt[3]{x^{2}} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}})}^{- 6}$

Step 2.3

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

${(\frac{\sqrt[3]{x^{2}} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}})}^{- 6}$

Step 2.4

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

${(\frac{\sqrt[3]{x^{2}} \sqrt{x}}{{\sqrt{x}}^{1 + 1}})}^{- 6}$

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

${(\frac{\sqrt[3]{x^{2}} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}})}^{- 6}$

Step 2.6.2

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

${(\frac{\sqrt[3]{x^{2}} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}})}^{- 6}$

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

${(\frac{\sqrt[3]{x^{2}} \sqrt{x}}{x^{1}})}^{- 6}$

Step 2.6.5

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

${(\frac{x^{\frac{2}{3}} \sqrt{x}}{x})}^{- 6}$

Step 3.1.2

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{4}{6}}$ .

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

Step 3.1.3

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

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

Step 3.1.4

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

${(\frac{\sqrt[6]{x^{4}} x^{\frac{1}{2}}}{x})}^{- 6}$

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.2

Combine using the product rule for radicals.

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

Step 3.3

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 3.3.1

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

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

Step 3.3.2

Add $4$ and $3$ .

${(\frac{\sqrt[6]{x^{7}}}{x})}^{- 6}$

Step 4

Simplify the numerator.

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Step 4.1

Factor out $x^{6}$ .

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

Step 4.2

Pull terms out from under the radical.

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

Step 5

Cancel the common factor of $x$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $\sqrt[6]{x}$ by $1$ .

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

Step 6

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

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

Step 7

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

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Step 7.1

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

$\frac{1}{{(x^{\frac{1}{6}})}^{6}}$

Step 7.2

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

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

Step 7.3

Combine $\frac{1}{6}$ and $6$ .

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

Step 7.4

Cancel the common factor of $6$ .

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Rewrite the expression.

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

Step 7.5

Simplify.

$\frac{1}{x}$