Precalculus Examples

$\frac{\sqrt{48 x^{3} y^{2}}}{\sqrt{4 x y^{3}}}$

Step 1

Combine $\sqrt{48 x^{3} y^{2}}$ and $\sqrt{4 x y^{3}}$ into a single radical.

$\sqrt{\frac{48 x^{3} y^{2}}{4 x y^{3}}}$

Step 2

Reduce the expression $\frac{48 x^{3} y^{2}}{4 x y^{3}}$ by cancelling the common factors.

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Step 2.1

Factor $4$ out of $48 x^{3} y^{2}$ .

$\sqrt{\frac{4 (12 x^{3} y^{2})}{4 x y^{3}}}$

Step 2.2

Factor $4$ out of $4 x y^{3}$ .

$\sqrt{\frac{4 (12 x^{3} y^{2})}{4 (x y^{3})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{4 (12 x^{3} y^{2})}{4 (x y^{3})}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{12 x^{3} y^{2}}{x y^{3}}}$

Step 3

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

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Step 3.1

Factor $x$ out of $12 x^{3} y^{2}$ .

$\sqrt{\frac{x (12 x^{2} y^{2})}{x y^{3}}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\sqrt{\frac{x (12 x^{2} y^{2})}{x (y^{3})}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{x (12 x^{2} y^{2})}{x y^{3}}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{12 x^{2} y^{2}}{y^{3}}}$

Step 4

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

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Step 4.1

Factor $y^{2}$ out of $12 x^{2} y^{2}$ .

$\sqrt{\frac{y^{2} (12 x^{2})}{y^{3}}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\sqrt{\frac{y^{2} (12 x^{2})}{y^{2} y}}$

Step 4.2.2

Cancel the common factor.

$\sqrt{\frac{y^{2} (12 x^{2})}{y^{2} y}}$

Step 4.2.3

Rewrite the expression.

$\sqrt{\frac{12 x^{2}}{y}}$

Step 5

Rewrite $\sqrt{\frac{12 x^{2}}{y}}$ as $\frac{\sqrt{12 x^{2}}}{\sqrt{y}}$ .

$\frac{\sqrt{12 x^{2}}}{\sqrt{y}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $12 x^{2}$ as ${(2 x)}^{2} \cdot 3$ .

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Step 6.1.1

Factor $4$ out of $12$ .

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

Step 6.1.2

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

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

Step 6.1.3

Move $3$ .

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

Step 6.1.4

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

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

Step 6.2

Pull terms out from under the radical.

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Simplify.

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

Step 9

Combine using the product rule for radicals.

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

Step 10

Move $3$ to the left of $y$ .

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