Simplify (a^(m/2)b^(n/3))^-6

${(a^{\frac{m}{2}} b^{\frac{n}{3}})}^{- 6}$

Step 1

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

$\frac{1}{{(a^{\frac{m}{2}} b^{\frac{n}{3}})}^{6}}$

Step 2

Simplify the denominator.

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

$\frac{1}{{(a^{\frac{m}{2}})}^{6} {(b^{\frac{n}{3}})}^{6}}$

Multiply the exponents in ${(a^{\frac{m}{2}})}^{6}$ .

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

$\frac{1}{a^{\frac{m}{2} \cdot 6} {(b^{\frac{n}{3}})}^{6}}$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\frac{1}{a^{\frac{m}{2} \cdot (2 (3))} {(b^{\frac{n}{3}})}^{6}}$

Cancel the common factor.

$\frac{1}{a^{\frac{m}{2} \cdot (2 \cdot 3)} {(b^{\frac{n}{3}})}^{6}}$

Rewrite the expression.

$\frac{1}{a^{m \cdot 3} {(b^{\frac{n}{3}})}^{6}}$

Move $3$ to the left of $m$ .

$\frac{1}{a^{3 m} {(b^{\frac{n}{3}})}^{6}}$

Multiply the exponents in ${(b^{\frac{n}{3}})}^{6}$ .

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

$\frac{1}{a^{3 m} b^{\frac{n}{3} \cdot 6}}$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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