Simplify ((a^2b^(1/3))/(a^3b))^(1/2)

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

Move $b^{\frac{1}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Multiply $b$ by $b^{- \frac{1}{3}}$ by adding the exponents.

Move $b^{- \frac{1}{3}}$ .

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

Multiply $b^{- \frac{1}{3}}$ by $b$ .

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

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $3$ .

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

Multiply by $1$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{1}{a b^{\frac{2}{3}}}$ .

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

Apply the product rule to $a b^{\frac{2}{3}}$ .

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

One to any power is one.

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

Multiply the exponents in ${(b^{\frac{2}{3}})}^{\frac{1}{2}}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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