Simplify ( square root of 20ab)/( square root of 45a^2b^3)

$\frac{\sqrt{20 a b}}{\sqrt{45 a^{2} b^{3}}}$

Step 1

Combine $\sqrt{20 a b}$ and $\sqrt{45 a^{2} b^{3}}$ into a single radical.

$\sqrt{\frac{20 a b}{45 a^{2} b^{3}}}$

Step 2

Reduce the expression $\frac{20 a b}{45 a^{2} b^{3}}$ by cancelling the common factors.

Factor $5$ out of $20 a b$ .

$\sqrt{\frac{5 (4 a b)}{45 a^{2} b^{3}}}$

Factor $5$ out of $45 a^{2} b^{3}$ .

$\sqrt{\frac{5 (4 a b)}{5 (9 a^{2} b^{3})}}$

Cancel the common factor.

$\sqrt{\frac{5 (4 a b)}{5 (9 a^{2} b^{3})}}$

Rewrite the expression.

$\sqrt{\frac{4 a b}{9 a^{2} b^{3}}}$

Step 3

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

Factor $a$ out of $4 a b$ .

$\sqrt{\frac{a (4 b)}{9 a^{2} b^{3}}}$

Cancel the common factors.

Factor $a$ out of $9 a^{2} b^{3}$ .

$\sqrt{\frac{a (4 b)}{a (9 a b^{3})}}$

Cancel the common factor.

$\sqrt{\frac{a (4 b)}{a (9 a b^{3})}}$

Rewrite the expression.

$\sqrt{\frac{4 b}{9 a b^{3}}}$

Step 4

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

Factor $b$ out of $4 b$ .

$\sqrt{\frac{b \cdot 4}{9 a b^{3}}}$

Cancel the common factors.

Factor $b$ out of $9 a b^{3}$ .

$\sqrt{\frac{b \cdot 4}{b (9 a b^{2})}}$

Cancel the common factor.

$\sqrt{\frac{b \cdot 4}{b (9 a b^{2})}}$

Rewrite the expression.

$\sqrt{\frac{4}{9 a b^{2}}}$

Step 5

Rewrite $\frac{4}{9 a b^{2}}$ as ${(\frac{2}{3 b})}^{2} \frac{1}{a}$ .

Factor the perfect power $2^{2}$ out of $4$ .

$\sqrt{\frac{2^{2} \cdot 1}{9 a b^{2}}}$

Factor the perfect power ${(3 b)}^{2}$ out of $9 a b^{2}$ .

$\sqrt{\frac{2^{2} \cdot 1}{{(3 b)}^{2} a}}$

Rearrange the fraction $\frac{2^{2} \cdot 1}{{(3 b)}^{2} a}$ .

$\sqrt{{(\frac{2}{3 b})}^{2} \frac{1}{a}}$

Step 6

Pull terms out from under the radical.

$\frac{2}{3 b} \sqrt{\frac{1}{a}}$

Step 7

Rewrite $\sqrt{\frac{1}{a}}$ as $\frac{\sqrt{1}}{\sqrt{a}}$ .

$\frac{2}{3 b} \cdot \frac{\sqrt{1}}{\sqrt{a}}$

Step 8

Any root of $1$ is $1$ .

$\frac{2}{3 b} \cdot \frac{1}{\sqrt{a}}$

Step 9

Multiply $\frac{1}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$\frac{2}{3 b} (\frac{1}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}})$

Step 10

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{\sqrt{a} \sqrt{a}}$

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

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{{\sqrt{a}}^{1} \sqrt{a}}$

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

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{{\sqrt{a}}^{1} {\sqrt{a}}^{1}}$

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

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{{\sqrt{a}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{{\sqrt{a}}^{2}}$

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

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

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{{(a^{\frac{1}{2}})}^{2}}$

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

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{a^{\frac{1}{2} \cdot 2}}$

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

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{a^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{a^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{a^{1}}$

Simplify.

$\frac{2}{3 b} \cdot \frac{\sqrt{a}}{a}$

Step 11

Multiply $\frac{2}{3 b}$ by $\frac{\sqrt{a}}{a}$ .

$\frac{2 \sqrt{a}}{3 b a}$

Simplify ( square root of 20ab)/( square root of 45a^2b^3)

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