Simplify (a^3b^(-1/2))^4

${(a^{3} b^{- \frac{1}{2}})}^{4}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Combine $a^{3}$ and $\frac{1}{b^{\frac{1}{2}}}$ .

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

Apply basic rules of exponents.

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

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

Multiply the exponents in ${(a^{3})}^{4}$ .

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

$\frac{a^{3 \cdot 4}}{{(b^{\frac{1}{2}})}^{4}}$

Multiply $3$ by $4$ .

$\frac{a^{12}}{{(b^{\frac{1}{2}})}^{4}}$

Multiply the exponents in ${(b^{\frac{1}{2}})}^{4}$ .

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

$\frac{a^{12}}{b^{\frac{1}{2} \cdot 4}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{a^{12}}{b^{\frac{1}{2} \cdot (2 (2))}}$

Cancel the common factor.

$\frac{a^{12}}{b^{\frac{1}{2} \cdot (2 \cdot 2)}}$

Rewrite the expression.

$\frac{a^{12}}{b^{2}}$