Algebra Examples

$\frac{4 x^{2} y^{- \frac{3}{2}}}{\sqrt[4]{81 x^{5} y^{4}}}$

Step 1

Move $y^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4 x^{2}}{\sqrt[4]{81 x^{5} y^{4}} y^{\frac{3}{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $81 x^{5} y^{4}$ as ${(3 x y)}^{4} x$ .

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Step 2.1.1

Rewrite $81$ as $3^{4}$ .

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

Step 2.1.2

Factor out $x^{4}$ .

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

Step 2.1.3

Move $x$ .

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

Step 2.1.4

Rewrite $3^{4} (x^{4}) y^{4}$ as ${(3 x y)}^{4}$ .

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

Step 2.2

Pull terms out from under the radical.

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

Step 2.3

Multiply $y$ by $y^{\frac{3}{2}}$ by adding the exponents.

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Step 2.3.1

Move $y^{\frac{3}{2}}$ .

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

Step 2.3.2

Multiply $y^{\frac{3}{2}}$ by $y$ .

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.3

Write $1$ as a fraction with a common denominator.

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

Step 2.3.4

Combine the numerators over the common denominator.

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

Step 2.3.5

Add $3$ and $2$ .

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

Step 3

Factor $x$ out of $4 x^{2}$ .

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

Step 4

Cancel the common factors.

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Step 4.1

Factor $x$ out of $3 x y^{\frac{5}{2}} \sqrt[4]{x}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

Move $\sqrt[4]{x}$ .

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $3$ .

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

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $4$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

$\frac{4 x {\sqrt[4]{x}}^{3}}{3 y^{\frac{5}{2}} x^{1}}$

Step 6.6.5

Simplify.

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

Step 7

Cancel the common factor.

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

Step 8

Rewrite the expression.

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

Step 9

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

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