Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $4 x^{5} y$ as ${(2 x^{2})}^{2} (x y)$ .

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Step 1.1.1

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

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

Step 1.1.2

Factor out $x^{4}$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

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

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Step 3.1

Move $x^{- \frac{1}{2}}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $2$ .

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

Step 3.6.2

Add $- 1$ and $4$ .

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

Move $\sqrt{x y}$ .

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Add $1$ and $1$ .

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

Step 5.7

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

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Step 5.7.1

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

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

Step 5.7.2

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

$\frac{y^{2} \sqrt{x y}}{2 x^{\frac{3}{2}} {(x y)}^{\frac{1}{2} \cdot 2}}$

Step 5.7.3

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

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

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

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

Step 5.7.4.2

Rewrite the expression.

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

Step 5.7.5

Simplify.

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

Step 6

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

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Step 6.1

Move $x$ .

$\frac{y^{2} \sqrt{x y}}{2 (x \cdot x^{\frac{3}{2}}) y}$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.3

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

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 6.5

Add $2$ and $3$ .

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

Step 7

Factor $y$ out of $y^{2} \sqrt{x y}$ .

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

Step 8

Cancel the common factors.

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Step 8.1

Factor $y$ out of $2 x^{\frac{5}{2}} y$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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