Simplify (a^(1/3)b^(-1/4))^-12

${(a^{\frac{1}{3}} b^{- \frac{1}{4}})}^{- 12}$

Step 1

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

${(a^{\frac{1}{3}} \frac{1}{b^{\frac{1}{4}}})}^{- 12}$

Step 2

Combine $a^{\frac{1}{3}}$ and $\frac{1}{b^{\frac{1}{4}}}$ .

${(\frac{a^{\frac{1}{3}}}{b^{\frac{1}{4}}})}^{- 12}$

Step 3

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{b^{\frac{1}{4}}}{a^{\frac{1}{3}}})}^{12}$

Step 4

Apply the product rule to $\frac{b^{\frac{1}{4}}}{a^{\frac{1}{3}}}$ .

$\frac{{(b^{\frac{1}{4}})}^{12}}{{(a^{\frac{1}{3}})}^{12}}$

Step 5

Multiply the exponents in ${(b^{\frac{1}{4}})}^{12}$ .

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

$\frac{b^{\frac{1}{4} \cdot 12}}{{(a^{\frac{1}{3}})}^{12}}$

Cancel the common factor of $4$ .

Factor $4$ out of $12$ .

$\frac{b^{\frac{1}{4} \cdot (4 (3))}}{{(a^{\frac{1}{3}})}^{12}}$

Cancel the common factor.

$\frac{b^{\frac{1}{4} \cdot (4 \cdot 3)}}{{(a^{\frac{1}{3}})}^{12}}$

Rewrite the expression.

$\frac{b^{3}}{{(a^{\frac{1}{3}})}^{12}}$

Step 6

Multiply the exponents in ${(a^{\frac{1}{3}})}^{12}$ .

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

$\frac{b^{3}}{a^{\frac{1}{3} \cdot 12}}$

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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