Algebra Examples

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

Step 1

Multiply $\frac{3 - \sqrt[4]{5 x^{2}}}{\sqrt[4]{3 x^{3}}}$ by $\frac{{\sqrt[4]{3 x^{3}}}^{3}}{{\sqrt[4]{3 x^{3}}}^{3}}$ .

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

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{3 - \sqrt[4]{5 x^{2}}}{\sqrt[4]{3 x^{3}}}$ by $\frac{{\sqrt[4]{3 x^{3}}}^{3}}{{\sqrt[4]{3 x^{3}}}^{3}}$ .

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

Step 2.2

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

$\frac{(3 - \sqrt[4]{5 x^{2}}) {\sqrt[4]{3 x^{3}}}^{3}}{{\sqrt[4]{3 x^{3}}}^{1} {\sqrt[4]{3 x^{3}}}^{3}}$

Step 2.3

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

$\frac{(3 - \sqrt[4]{5 x^{2}}) {\sqrt[4]{3 x^{3}}}^{3}}{{\sqrt[4]{3 x^{3}}}^{1 + 3}}$

Step 2.4

Add $1$ and $3$ .

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

Step 2.5

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

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3 x^{3}}$ as ${(3 x^{3})}^{\frac{1}{4}}$ .

$\frac{(3 - \sqrt[4]{5 x^{2}}) {\sqrt[4]{3 x^{3}}}^{3}}{{({(3 x^{3})}^{\frac{1}{4}})}^{4}}$

Step 2.5.2

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

$\frac{(3 - \sqrt[4]{5 x^{2}}) {\sqrt[4]{3 x^{3}}}^{3}}{{(3 x^{3})}^{\frac{1}{4} \cdot 4}}$

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $4$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

$\frac{(3 - \sqrt[4]{5 x^{2}}) {\sqrt[4]{3 x^{3}}}^{3}}{{(3 x^{3})}^{1}}$

Step 2.5.5

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

Apply the product rule to $3 x^{3}$ .

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

Step 3.3

Raise $3$ to the power of $3$ .

$\frac{(3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{27 {(x^{3})}^{3}}}{3 x^{3}}$

Step 3.4

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

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Step 3.4.1

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

$\frac{(3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{27 x^{3 \cdot 3}}}{3 x^{3}}$

Step 3.4.2

Multiply $3$ by $3$ .

$\frac{(3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{27 x^{9}}}{3 x^{3}}$

Step 3.5

Rewrite $27 x^{9}$ as ${(x^{2})}^{4} \cdot (27 x)$ .

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Step 3.5.1

Factor out $x^{8}$ .

$\frac{(3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{27 (x^{8} x)}}{3 x^{3}}$

Step 3.5.2

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

$\frac{(3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{27 ({(x^{2})}^{4} x)}}{3 x^{3}}$

Step 3.5.3

Reorder $27$ and ${(x^{2})}^{4}$ .

$\frac{(3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{{(x^{2})}^{4} \cdot 27 x}}{3 x^{3}}$

Step 3.5.4

Add parentheses.

$\frac{(3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{{(x^{2})}^{4} \cdot (27 x)}}{3 x^{3}}$

Step 3.6

Pull terms out from under the radical.

$\frac{(3 - \sqrt[4]{5 x^{2}}) x^{2} \sqrt[4]{27 x}}{3 x^{3}}$

Step 4

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

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Step 4.1

Factor $x^{2}$ out of $(3 - \sqrt[4]{5 x^{2}}) x^{2} \sqrt[4]{27 x}$ .

$\frac{x^{2} ((3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{27 x})}{3 x^{3}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\frac{x^{2} ((3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{27 x})}{x^{2} (3 x)}$

Step 4.2.2

Cancel the common factor.

$\frac{x^{2} ((3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{27 x})}{x^{2} (3 x)}$

Step 4.2.3

Rewrite the expression.

$\frac{(3 - \sqrt[4]{5 x^{2}}) \sqrt[4]{27 x}}{3 x}$

Step 5

Apply the distributive property.

$\frac{3 \sqrt[4]{27 x} - \sqrt[4]{5 x^{2}} \sqrt[4]{27 x}}{3 x}$

Step 6

Multiply $- \sqrt[4]{5 x^{2}} \sqrt[4]{27 x}$ .

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Step 6.1

Combine using the product rule for radicals.

$\frac{3 \sqrt[4]{27 x} - \sqrt[4]{27 x (5 x^{2})}}{3 x}$

Step 6.2

Multiply $5$ by $27$ .

$\frac{3 \sqrt[4]{27 x} - \sqrt[4]{135 x \cdot x^{2}}}{3 x}$

Step 6.3

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

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Step 6.3.1

Move $x^{2}$ .

$\frac{3 \sqrt[4]{27 x} - \sqrt[4]{135 (x^{2} x)}}{3 x}$

Step 6.3.2

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

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Step 6.3.2.1

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

$\frac{3 \sqrt[4]{27 x} - \sqrt[4]{135 (x^{2} x^{1})}}{3 x}$

Step 6.3.2.2

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

$\frac{3 \sqrt[4]{27 x} - \sqrt[4]{135 x^{2 + 1}}}{3 x}$

Step 6.3.3

Add $2$ and $1$ .

$\frac{3 \sqrt[4]{27 x} - \sqrt[4]{135 x^{3}}}{3 x}$

Step 7

Split the fraction $\frac{3 \sqrt[4]{27 x} - \sqrt[4]{135 x^{3}}}{3 x}$ into two fractions.

$\frac{3 \sqrt[4]{27 x}}{3 x} + \frac{- \sqrt[4]{135 x^{3}}}{3 x}$

Step 8

Cancel the common factor of $3$ .

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Step 8.1

Cancel the common factor.

$\frac{3 \sqrt[4]{27 x}}{3 x} + \frac{- \sqrt[4]{135 x^{3}}}{3 x}$

Step 8.2

Rewrite the expression.

$\frac{\sqrt[4]{27 x}}{x} + \frac{- \sqrt[4]{135 x^{3}}}{3 x}$

Step 9

Move the negative in front of the fraction.

$\frac{\sqrt[4]{27 x}}{x} - \frac{\sqrt[4]{135 x^{3}}}{3 x}$