Simplify (-2ab^2)^2(-ab)^3

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

Step 1

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

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

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

Apply the product rule to $- 2 a$ .

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

Step 2

Raise $- 2$ to the power of $2$ .

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

Step 3

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

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

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

Multiply $2$ by $2$ .

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

Step 4

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

Apply the product rule to $- a b$ .

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

Apply the product rule to $- a$ .

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

Step 5

Multiply $a^{2}$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

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

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

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

Add $3$ and $2$ .

$4 a^{5} b^{4} ({(- 1)}^{3} b^{3})$

Step 6

Multiply $b^{4}$ by $b^{3}$ by adding the exponents.

Move $b^{3}$ .

$4 a^{5} (b^{3} b^{4}) {(- 1)}^{3}$

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

$4 a^{5} b^{3 + 4} {(- 1)}^{3}$

Add $3$ and $4$ .

$4 a^{5} b^{7} {(- 1)}^{3}$

Step 7

Raise $- 1$ to the power of $3$ .

$4 a^{5} b^{7} \cdot - 1$

Step 8

Multiply $- 1$ by $4$ .

$- 4 a^{5} b^{7}$