Simplify the Radical Expression fourth root of (4a^3b^2)/(2a^2b^3)

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

Step 1

Reduce the expression $\frac{4 a^{3} b^{2}}{2 a^{2} b^{3}}$ by cancelling the common factors.

Factor $2$ out of $4 a^{3} b^{2}$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\sqrt[4]{\frac{2 a}{b}}$

Step 4

Rewrite $\sqrt[4]{\frac{2 a}{b}}$ as $\frac{\sqrt[4]{2 a}}{\sqrt[4]{b}}$ .

$\frac{\sqrt[4]{2 a}}{\sqrt[4]{b}}$

Step 5

Multiply $\frac{\sqrt[4]{2 a}}{\sqrt[4]{b}}$ by $\frac{{\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{3}}$ .

$\frac{\sqrt[4]{2 a}}{\sqrt[4]{b}} \cdot \frac{{\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{3}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{2 a}}{\sqrt[4]{b}}$ by $\frac{{\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{3}}$ .

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

Raise $\sqrt[4]{b}$ to the power of $1$ .

$\frac{\sqrt[4]{2 a} {\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{1} {\sqrt[4]{b}}^{3}}$

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

$\frac{\sqrt[4]{2 a} {\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{1 + 3}}$

Add $1$ and $3$ .

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

Rewrite ${\sqrt[4]{b}}^{4}$ as $b$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{b}$ as $b^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{2 a} {\sqrt[4]{b}}^{3}}{{(b^{\frac{1}{4}})}^{4}}$

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

$\frac{\sqrt[4]{2 a} {\sqrt[4]{b}}^{3}}{b^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt[4]{2 a} {\sqrt[4]{b}}^{3}}{b^{1}}$

Simplify.

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

Step 7

Rewrite ${\sqrt[4]{b}}^{3}$ as ${(b^{3})}^{\frac{1}{4}}$ .

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

Step 8

Combine using the product rule for radicals.

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

Simplify the Radical Expression fourth root of (4a^3b^2)/(2a^2b^3)

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