Simplify (-3a^3)^2(2ab)^4

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

Step 1

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

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

Step 2

Raise $- 3$ to the power of $2$ .

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

Step 3

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

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

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

Multiply $3$ by $2$ .

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

Step 4

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

Apply the product rule to $2 a b$ .

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

Apply the product rule to $2 a$ .

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

Step 5

Multiply $a^{6}$ by $a^{4}$ by adding the exponents.

Move $a^{4}$ .

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

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

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

Add $4$ and $6$ .

$9 a^{10} (2^{4} b^{4})$

Step 6

Rewrite using the commutative property of multiplication.

$9 \cdot 2^{4} a^{10} b^{4}$

Step 7

Raise $2$ to the power of $4$ .

$9 \cdot 16 a^{10} b^{4}$

Step 8

Multiply $9$ by $16$ .

$144 a^{10} b^{4}$