Simplify (-a^2b^3)^2*(4a^3b^2)+9a^7b^8

${(- a^{2} b^{3})}^{2} \cdot (4 a^{3} b^{2}) + 9 a^{7} b^{8}$

Step 1

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 {(- a^{2} b^{3})}^{2} a^{3} b^{2} + 9 a^{7} b^{8}$

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

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

$4 ({(- a^{2})}^{2} {(b^{3})}^{2}) a^{3} b^{2} + 9 a^{7} b^{8}$

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

$4 ({(- 1)}^{2} {(a^{2})}^{2} {(b^{3})}^{2}) a^{3} b^{2} + 9 a^{7} b^{8}$

Raise $- 1$ to the power of $2$ .

$4 (1 {(a^{2})}^{2} {(b^{3})}^{2}) a^{3} b^{2} + 9 a^{7} b^{8}$

Multiply ${(a^{2})}^{2}$ by $1$ .

$4 ({(a^{2})}^{2} {(b^{3})}^{2}) a^{3} b^{2} + 9 a^{7} b^{8}$

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

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

$4 (a^{2 \cdot 2} {(b^{3})}^{2}) a^{3} b^{2} + 9 a^{7} b^{8}$

Multiply $2$ by $2$ .

$4 (a^{4} {(b^{3})}^{2}) a^{3} b^{2} + 9 a^{7} b^{8}$

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

Move $a^{3}$ .

$4 (a^{3} a^{4} {(b^{3})}^{2}) b^{2} + 9 a^{7} b^{8}$

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

$4 (a^{3 + 4} {(b^{3})}^{2}) b^{2} + 9 a^{7} b^{8}$

Add $3$ and $4$ .

$4 (a^{7} {(b^{3})}^{2}) b^{2} + 9 a^{7} b^{8}$

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

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

$4 (a^{7} b^{3 \cdot 2}) b^{2} + 9 a^{7} b^{8}$

Multiply $3$ by $2$ .

$4 (a^{7} b^{6}) b^{2} + 9 a^{7} b^{8}$

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

Move $b^{2}$ .

$4 (a^{7} (b^{2} b^{6})) + 9 a^{7} b^{8}$

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

$4 (a^{7} b^{2 + 6}) + 9 a^{7} b^{8}$

Add $2$ and $6$ .

$4 (a^{7} b^{8}) + 9 a^{7} b^{8}$

$4 a^{7} b^{8} + 9 a^{7} b^{8}$

Step 2

Add $4 a^{7} b^{8}$ and $9 a^{7} b^{8}$ .

$13 a^{7} b^{8}$