Finite Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor out $x^{3}$ .

$\frac{\sqrt[3]{x^{3} x^{2}}}{\sqrt[4]{x^{5}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{x \sqrt[3]{x^{2}}}{\sqrt[4]{x^{5}}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor out $x^{4}$ .

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

Cancel the common factor of $x$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $3$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

Cancel the common factor of $4$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Simplify.

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

Step 6

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

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

Step 7

Simplify the numerator.

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Step 7.1

Rewrite the expression using the least common index of $12$ .

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Step 7.1.1

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

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

Step 7.1.2

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{8}{12}}$ .

$\frac{x^{\frac{8}{12}} \sqrt[4]{x^{3}}}{x}$

Step 7.1.3

Rewrite $x^{\frac{8}{12}}$ as $\sqrt[12]{x^{8}}$ .

$\frac{\sqrt[12]{x^{8}} \sqrt[4]{x^{3}}}{x}$

Step 7.1.4

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

$\frac{\sqrt[12]{x^{8}} x^{\frac{3}{4}}}{x}$

Step 7.1.5

Rewrite $x^{\frac{3}{4}}$ as $x^{\frac{9}{12}}$ .

$\frac{\sqrt[12]{x^{8}} x^{\frac{9}{12}}}{x}$

Step 7.1.6

Rewrite $x^{\frac{9}{12}}$ as $\sqrt[12]{x^{9}}$ .

$\frac{\sqrt[12]{x^{8}} \sqrt[12]{x^{9}}}{x}$

Step 7.2

Combine using the product rule for radicals.

$\frac{\sqrt[12]{x^{8} x^{9}}}{x}$

Step 7.3

Multiply $x^{8}$ by $x^{9}$ by adding the exponents.

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Step 7.3.1

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

$\frac{\sqrt[12]{x^{8 + 9}}}{x}$

Step 7.3.2

Add $8$ and $9$ .

$\frac{\sqrt[12]{x^{17}}}{x}$

Step 8

Simplify the numerator.

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Step 8.1

Factor out $x^{12}$ .

$\frac{\sqrt[12]{x^{12} x^{5}}}{x}$

Step 8.2

Pull terms out from under the radical.

$\frac{x \sqrt[12]{x^{5}}}{x}$

Step 9

Cancel the common factor of $x$ .

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Step 9.1

Cancel the common factor.

$\frac{x \sqrt[12]{x^{5}}}{x}$

Step 9.2

Divide $\sqrt[12]{x^{5}}$ by $1$ .

$\sqrt[12]{x^{5}}$