Algebra Examples

$\frac{\sqrt{36 x^{5} y^{2}}}{4 x^{\frac{1}{2}} y^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $36 x^{5} y^{2}$ as ${(6 x^{2} y)}^{2} x$ .

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Step 1.1.1

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2} x^{5} y^{2}}}{4 x^{\frac{1}{2}} y^{4}}$

Step 1.1.2

Factor out $x^{4}$ .

$\frac{\sqrt{6^{2} (x^{4} x) y^{2}}}{4 x^{\frac{1}{2}} y^{4}}$

Step 1.1.3

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

$\frac{\sqrt{6^{2} ({(x^{2})}^{2} x) y^{2}}}{4 x^{\frac{1}{2}} y^{4}}$

Step 1.1.4

Move $x$ .

$\frac{\sqrt{6^{2} {(x^{2})}^{2} y^{2} x}}{4 x^{\frac{1}{2}} y^{4}}$

Step 1.1.5

Rewrite $6^{2} ({(x^{2})}^{2}) y^{2}$ as ${(6 x^{2} y)}^{2}$ .

$\frac{\sqrt{{(6 x^{2} y)}^{2} x}}{4 x^{\frac{1}{2}} y^{4}}$

Step 1.2

Pull terms out from under the radical.

$\frac{6 x^{2} y \sqrt{x}}{4 x^{\frac{1}{2}} y^{4}}$

Step 2

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

$\frac{6 x^{2} y \sqrt{x} x^{- \frac{1}{2}}}{4 y^{4}}$

Step 3

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

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Step 3.1

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

$\frac{6 (x^{- \frac{1}{2}} x^{2}) y \sqrt{x}}{4 y^{4}}$

Step 3.2

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

$\frac{6 x^{- \frac{1}{2} + 2} y \sqrt{x}}{4 y^{4}}$

Step 3.3

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

$\frac{6 x^{- \frac{1}{2} + 2 \cdot \frac{2}{2}} y \sqrt{x}}{4 y^{4}}$

Step 3.4

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

$\frac{6 x^{- \frac{1}{2} + \frac{2 \cdot 2}{2}} y \sqrt{x}}{4 y^{4}}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{6 x^{\frac{- 1 + 2 \cdot 2}{2}} y \sqrt{x}}{4 y^{4}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $2$ .

$\frac{6 x^{\frac{- 1 + 4}{2}} y \sqrt{x}}{4 y^{4}}$

Step 3.6.2

Add $- 1$ and $4$ .

$\frac{6 x^{\frac{3}{2}} y \sqrt{x}}{4 y^{4}}$

Step 4

Factor $2$ out of $6 x^{\frac{3}{2}} y \sqrt{x}$ .

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

Step 5

Cancel the common factors.

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Step 5.1

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

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Factor $y$ out of $3 x^{\frac{3}{2}} y \sqrt{x}$ .

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

Step 7

Cancel the common factors.

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Combine the numerators over the common denominator.

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

Step 8.4

Add $1$ and $3$ .

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

Step 8.5

Cancel the common factor of $4$ and $2$ .

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Step 8.5.1

Factor $2$ out of $4$ .

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

Step 8.5.2

Cancel the common factors.

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Step 8.5.2.1

Factor $2$ out of $2$ .

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

Step 8.5.2.2

Cancel the common factor.

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

Step 8.5.2.3

Rewrite the expression.

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

Step 8.5.2.4

Divide $2$ by $1$ .

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