Simplify square root of (4ab^5)/(8a^4b^2)

$\sqrt{\frac{4 a b^{5}}{8 a^{4} b^{2}}}$

Reduce the expression $\frac{4 a b^{5}}{8 a^{4} b^{2}}$ by cancelling the common factors.

Factor $4$ out of $4 a b^{5}$ .

$\sqrt{\frac{4 (a b^{5})}{8 a^{4} b^{2}}}$

Factor $4$ out of $8 a^{4} b^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{\frac{a b^{5}}{2 a^{4} b^{2}}}$

Cancel the common factor of $a$ and $a^{4}$ .

Factor $a$ out of $a b^{5}$ .

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

Cancel the common factors.

Factor $a$ out of $2 a^{4} b^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{\frac{b^{5}}{2 a^{3} b^{2}}}$

Cancel the common factor of $b^{5}$ and $b^{2}$ .

Factor $b^{2}$ out of $b^{5}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor the perfect power $b^{2}$ out of $b^{3}$ .

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

Factor the perfect power $a^{2}$ out of $2 a^{3}$ .

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

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

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

Pull terms out from under the radical.

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

Rewrite $\sqrt{\frac{b}{2 a}}$ as $\frac{\sqrt{b}}{\sqrt{2 a}}$ .

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

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt{2 a}$ to the power of $1$ .

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

Raise $\sqrt{2 a}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Combine using the product rule for radicals.

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

Multiply $\frac{b}{a} \cdot \frac{\sqrt{b \cdot 2 a}}{2 a}$ .

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Move $2$ to the left of $b$ .

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

Simplify square root of (4ab^5)/(8a^4b^2)

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