Simplify ((a^2b^3)^2)/((ab)^-2)

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

Step 1

Move ${(a b)}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(a^{2} b^{3})}^{2} {(a b)}^{2}$

Step 2

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

${(a^{2})}^{2} {(b^{3})}^{2} {(a b)}^{2}$

Step 3

Multiply the exponents in ${(a^{2})}^{2}$ .

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

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

Multiply $2$ by $2$ .

$a^{4} {(b^{3})}^{2} {(a b)}^{2}$

Step 4

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

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

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

Multiply $3$ by $2$ .

$a^{4} b^{6} {(a b)}^{2}$

Step 5

Apply the product rule to $a b$ .

$a^{4} b^{6} (a^{2} b^{2})$

Step 6

Multiply $a^{4}$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

$a^{2} a^{4} b^{6} b^{2}$

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

$a^{2 + 4} b^{6} b^{2}$

Add $2$ and $4$ .

$a^{6} b^{6} b^{2}$

Step 7

Multiply $b^{6}$ by $b^{2}$ by adding the exponents.

Move $b^{2}$ .

$a^{6} (b^{2} b^{6})$

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

$a^{6} b^{2 + 6}$

Add $2$ and $6$ .

$a^{6} b^{8}$