Evaluate (3a^3b^3)(4ab^2)^2

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

Step 1

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

Apply the product rule to $4 a b^{2}$ .

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

Apply the product rule to $4 a$ .

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

Step 2

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

Move $a^{2}$ .

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

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

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

Add $2$ and $3$ .

$3 a^{5} b^{3} (4^{2} {(b^{2})}^{2})$

Step 3

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

$3 a^{5} b^{3} (16 {(b^{2})}^{2})$

Step 4

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

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

$3 a^{5} b^{3} (16 b^{2 \cdot 2})$

Multiply $2$ by $2$ .

$3 a^{5} b^{3} (16 b^{4})$

Step 5

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

Move $b^{4}$ .

$3 a^{5} (b^{4} b^{3}) \cdot 16$

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

$3 a^{5} b^{4 + 3} \cdot 16$

Add $4$ and $3$ .

$3 a^{5} b^{7} \cdot 16$

Step 6

Multiply $16$ by $3$ .

$48 a^{5} b^{7}$