Algebra Examples

$\frac{2 \sqrt{108}}{\sqrt{180 y}}$

Step 1

Combine $\sqrt{108}$ and $\sqrt{180 y}$ into a single radical.

$2 \sqrt{\frac{108}{180 y}}$

Step 2

Reduce the expression $\frac{108}{180 y}$ by cancelling the common factors.

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Step 2.1

Factor $36$ out of $108$ .

$2 \sqrt{\frac{36 (3)}{180 y}}$

Step 2.2

Factor $36$ out of $180 y$ .

$2 \sqrt{\frac{36 (3)}{36 (5 y)}}$

Step 2.3

Cancel the common factor.

$2 \sqrt{\frac{36 \cdot 3}{36 (5 y)}}$

Step 2.4

Rewrite the expression.

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

Step 3

Rewrite $\sqrt{\frac{3}{5 y}}$ as $\frac{\sqrt{3}}{\sqrt{5 y}}$ .

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

Step 4

Multiply $\frac{\sqrt{3}}{\sqrt{5 y}}$ by $\frac{\sqrt{5 y}}{\sqrt{5 y}}$ .

$2 (\frac{\sqrt{3}}{\sqrt{5 y}} \cdot \frac{\sqrt{5 y}}{\sqrt{5 y}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt{3}}{\sqrt{5 y}}$ by $\frac{\sqrt{5 y}}{\sqrt{5 y}}$ .

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

Step 5.2

Raise $\sqrt{5 y}$ to the power of $1$ .

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

Step 5.3

Raise $\sqrt{5 y}$ to the power of $1$ .

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

Rewrite ${\sqrt{5 y}}^{2}$ as $5 y$ .

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Step 5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 y}$ as ${(5 y)}^{\frac{1}{2}}$ .

$2 \frac{\sqrt{3} \sqrt{5 y}}{{({(5 y)}^{\frac{1}{2}})}^{2}}$

Step 5.6.2

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

$2 \frac{\sqrt{3} \sqrt{5 y}}{{(5 y)}^{\frac{1}{2} \cdot 2}}$

Step 5.6.3

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

$2 \frac{\sqrt{3} \sqrt{5 y}}{{(5 y)}^{\frac{2}{2}}}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$2 \frac{\sqrt{3} \sqrt{5 y}}{{(5 y)}^{\frac{2}{2}}}$

Step 5.6.4.2

Rewrite the expression.

$2 \frac{\sqrt{3} \sqrt{5 y}}{{(5 y)}^{1}}$

Step 5.6.5

Simplify.

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

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

$2 \frac{\sqrt{3 (5 y)}}{5 y}$

Step 6.2

Multiply $3$ by $5$ .

$2 \frac{\sqrt{15 y}}{5 y}$

Step 7

Combine $2$ and $\frac{\sqrt{15 y}}{5 y}$ .

$\frac{2 \sqrt{15 y}}{5 y}$