Algebra Examples

$\frac{4 x^{- 2} y^{\frac{7}{2}}}{\sqrt{16 x y^{3}}}$

Step 1

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

$\frac{4 y^{\frac{7}{2}}}{\sqrt{16 x y^{3}} x^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16 x y^{3}$ as ${(4 y)}^{2} (x y)$ .

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Step 2.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{4 y^{\frac{7}{2}}}{\sqrt{4^{2} x y^{3}} x^{2}}$

Step 2.1.2

Factor out $y^{2}$ .

$\frac{4 y^{\frac{7}{2}}}{\sqrt{4^{2} x (y^{2} y)} x^{2}}$

Step 2.1.3

Move $x$ .

$\frac{4 y^{\frac{7}{2}}}{\sqrt{4^{2} y^{2} x y} x^{2}}$

Step 2.1.4

Rewrite $4^{2} y^{2}$ as ${(4 y)}^{2}$ .

$\frac{4 y^{\frac{7}{2}}}{\sqrt{{(4 y)}^{2} x y} x^{2}}$

Step 2.1.5

Add parentheses.

$\frac{4 y^{\frac{7}{2}}}{\sqrt{{(4 y)}^{2} (x y)} x^{2}}$

Step 2.2

Pull terms out from under the radical.

$\frac{4 y^{\frac{7}{2}}}{4 y \sqrt{x y} x^{2}}$

Step 3

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

$\frac{4 y^{\frac{7}{2}} y^{- 1}}{4 \sqrt{x y} x^{2}}$

Step 4

Multiply $y^{\frac{7}{2}}$ by $y^{- 1}$ by adding the exponents.

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Step 4.1

Move $y^{- 1}$ .

$\frac{4 (y^{- 1} y^{\frac{7}{2}})}{4 \sqrt{x y} x^{2}}$

Step 4.2

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

$\frac{4 y^{- 1 + \frac{7}{2}}}{4 \sqrt{x y} x^{2}}$

Step 4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 y^{- 1 \cdot \frac{2}{2} + \frac{7}{2}}}{4 \sqrt{x y} x^{2}}$

Step 4.4

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

$\frac{4 y^{\frac{- 1 \cdot 2}{2} + \frac{7}{2}}}{4 \sqrt{x y} x^{2}}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{4 y^{\frac{- 1 \cdot 2 + 7}{2}}}{4 \sqrt{x y} x^{2}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

$\frac{4 y^{\frac{- 2 + 7}{2}}}{4 \sqrt{x y} x^{2}}$

Step 4.6.2

Add $- 2$ and $7$ .

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

Step 5

Cancel the common factor.

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

Step 6

Rewrite the expression.

$\frac{y^{\frac{5}{2}}}{\sqrt{x y} x^{2}}$

Step 7

Multiply $\frac{y^{\frac{5}{2}}}{\sqrt{x y} x^{2}}$ by $\frac{\sqrt{x y}}{\sqrt{x y}}$ .

$\frac{y^{\frac{5}{2}}}{\sqrt{x y} x^{2}} \cdot \frac{\sqrt{x y}}{\sqrt{x y}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{y^{\frac{5}{2}}}{\sqrt{x y} x^{2}}$ by $\frac{\sqrt{x y}}{\sqrt{x y}}$ .

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{\sqrt{x y} x^{2} \sqrt{x y}}$

Step 8.2

Move $\sqrt{x y}$ .

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} (\sqrt{x y} \sqrt{x y})}$

Step 8.3

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

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} ({\sqrt{x y}}^{1} \sqrt{x y})}$

Step 8.4

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

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} ({\sqrt{x y}}^{1} {\sqrt{x y}}^{1})}$

Step 8.5

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

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} {\sqrt{x y}}^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} {\sqrt{x y}}^{2}}$

Step 8.7

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

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Step 8.7.1

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

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} {({(x y)}^{\frac{1}{2}})}^{2}}$

Step 8.7.2

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

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} {(x y)}^{\frac{1}{2} \cdot 2}}$

Step 8.7.3

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

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} {(x y)}^{\frac{2}{2}}}$

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} {(x y)}^{\frac{2}{2}}}$

Step 8.7.4.2

Rewrite the expression.

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} {(x y)}^{1}}$

Step 8.7.5

Simplify.

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{2} (x y)}$

Step 9

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

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Step 9.1

Move $x$ .

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x \cdot x^{2} y}$

Step 9.2

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

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Step 9.2.1

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

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{1} x^{2} y}$

Step 9.2.2

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

$\frac{y^{\frac{5}{2}} \sqrt{x y}}{x^{1 + 2} y}$

Step 9.3

Add $1$ and $2$ .

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

Step 10

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

$\frac{y^{\frac{5}{2}} \sqrt{x y} y^{- 1}}{x^{3}}$

Step 11

Multiply $y^{\frac{5}{2}}$ by $y^{- 1}$ by adding the exponents.

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Step 11.1

Move $y^{- 1}$ .

$\frac{y^{- 1} y^{\frac{5}{2}} \sqrt{x y}}{x^{3}}$

Step 11.2

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

$\frac{y^{- 1 + \frac{5}{2}} \sqrt{x y}}{x^{3}}$

Step 11.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y^{- 1 \cdot \frac{2}{2} + \frac{5}{2}} \sqrt{x y}}{x^{3}}$

Step 11.4

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

$\frac{y^{\frac{- 1 \cdot 2}{2} + \frac{5}{2}} \sqrt{x y}}{x^{3}}$

Step 11.5

Combine the numerators over the common denominator.

$\frac{y^{\frac{- 1 \cdot 2 + 5}{2}} \sqrt{x y}}{x^{3}}$

Step 11.6

Simplify the numerator.

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Step 11.6.1

Multiply $- 1$ by $2$ .

$\frac{y^{\frac{- 2 + 5}{2}} \sqrt{x y}}{x^{3}}$

Step 11.6.2

Add $- 2$ and $5$ .

$\frac{y^{\frac{3}{2}} \sqrt{x y}}{x^{3}}$