Pre-Algebra Examples

$\frac{\sqrt[7]{x^{4}}}{\sqrt[8]{x}}$

Step 1

Multiply $\frac{\sqrt[7]{x^{4}}}{\sqrt[8]{x}}$ by $\frac{{\sqrt[8]{x}}^{7}}{{\sqrt[8]{x}}^{7}}$ .

$\frac{\sqrt[7]{x^{4}}}{\sqrt[8]{x}} \cdot \frac{{\sqrt[8]{x}}^{7}}{{\sqrt[8]{x}}^{7}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[7]{x^{4}}}{\sqrt[8]{x}}$ by $\frac{{\sqrt[8]{x}}^{7}}{{\sqrt[8]{x}}^{7}}$ .

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{\sqrt[8]{x} {\sqrt[8]{x}}^{7}}$

Step 2.2

Raise $\sqrt[8]{x}$ to the power of $1$ .

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{{\sqrt[8]{x}}^{1} {\sqrt[8]{x}}^{7}}$

Step 2.3

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

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{{\sqrt[8]{x}}^{1 + 7}}$

Step 2.4

Add $1$ and $7$ .

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{{\sqrt[8]{x}}^{8}}$

Step 2.5

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

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{x}$ as $x^{\frac{1}{8}}$ .

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{{(x^{\frac{1}{8}})}^{8}}$

Step 2.5.2

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

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{x^{\frac{1}{8} \cdot 8}}$

Step 2.5.3

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

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{x^{\frac{8}{8}}}$

Step 2.5.4

Cancel the common factor of $8$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{x^{\frac{8}{8}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{x^{1}}$

Step 2.5.5

Simplify.

$\frac{\sqrt[7]{x^{4}} {\sqrt[8]{x}}^{7}}{x}$

Step 3

Rewrite ${\sqrt[8]{x}}^{7}$ as $\sqrt[8]{x^{7}}$ .

$\frac{\sqrt[7]{x^{4}} \sqrt[8]{x^{7}}}{x}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $56$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{4}}$ as $x^{\frac{4}{7}}$ .

$\frac{x^{\frac{4}{7}} \sqrt[8]{x^{7}}}{x}$

Step 4.1.2

Rewrite $x^{\frac{4}{7}}$ as $x^{\frac{32}{56}}$ .

$\frac{x^{\frac{32}{56}} \sqrt[8]{x^{7}}}{x}$

Step 4.1.3

Rewrite $x^{\frac{32}{56}}$ as $\sqrt[56]{x^{32}}$ .

$\frac{\sqrt[56]{x^{32}} \sqrt[8]{x^{7}}}{x}$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{x^{7}}$ as $x^{\frac{7}{8}}$ .

$\frac{\sqrt[56]{x^{32}} x^{\frac{7}{8}}}{x}$

Step 4.1.5

Rewrite $x^{\frac{7}{8}}$ as $x^{\frac{49}{56}}$ .

$\frac{\sqrt[56]{x^{32}} x^{\frac{49}{56}}}{x}$

Step 4.1.6

Rewrite $x^{\frac{49}{56}}$ as $\sqrt[56]{x^{49}}$ .

$\frac{\sqrt[56]{x^{32}} \sqrt[56]{x^{49}}}{x}$

Step 4.2

Combine using the product rule for radicals.

$\frac{\sqrt[56]{x^{32} x^{49}}}{x}$

Step 4.3

Multiply $x^{32}$ by $x^{49}$ by adding the exponents.

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Step 4.3.1

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

$\frac{\sqrt[56]{x^{32 + 49}}}{x}$

Step 4.3.2

Add $32$ and $49$ .

$\frac{\sqrt[56]{x^{81}}}{x}$

Step 5

Simplify the numerator.

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Step 5.1

Factor out $x^{56}$ .

$\frac{\sqrt[56]{x^{56} x^{25}}}{x}$

Step 5.2

Pull terms out from under the radical.

$\frac{x \sqrt[56]{x^{25}}}{x}$

Step 6

Cancel the common factor of $x$ .

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Step 6.1

Cancel the common factor.

$\frac{x \sqrt[56]{x^{25}}}{x}$

Step 6.2

Divide $\sqrt[56]{x^{25}}$ by $1$ .

$\sqrt[56]{x^{25}}$