Simplify (-4a^3b^2)^2*1/2a^3b

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

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

Apply the product rule to $- 4 a^{3} b^{2}$ .

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

Apply the product rule to $- 4 a^{3}$ .

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

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

$16 {(a^{3})}^{2} {(b^{2})}^{2} \cdot \frac{1}{2} \cdot (a^{3} b)$

Cancel the common factor of $2$ .

Factor $2$ out of $16 {(a^{3})}^{2} {(b^{2})}^{2}$ .

$2 (8 {(a^{3})}^{2} {(b^{2})}^{2}) \cdot \frac{1}{2} \cdot (a^{3} b)$

Cancel the common factor.

$2 (8 {(a^{3})}^{2} {(b^{2})}^{2}) \cdot \frac{1}{2} \cdot (a^{3} b)$

Rewrite the expression.

$8 {(a^{3})}^{2} {(b^{2})}^{2} \cdot (a^{3} b)$

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

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

$8 a^{3 \cdot 2} {(b^{2})}^{2} \cdot (a^{3} b)$

Multiply $3$ by $2$ .

$8 a^{6} {(b^{2})}^{2} \cdot (a^{3} b)$

Multiply $a^{6}$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

$8 (a^{3} a^{6}) {(b^{2})}^{2} \cdot b$

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

$8 a^{3 + 6} {(b^{2})}^{2} \cdot b$

Add $3$ and $6$ .

$8 a^{9} {(b^{2})}^{2} \cdot b$

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

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

$8 a^{9} b^{2 \cdot 2} \cdot b$

Multiply $2$ by $2$ .

$8 a^{9} b^{4} \cdot b$

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

Move $b$ .

$8 a^{9} (b \cdot b^{4})$

Multiply $b$ by $b^{4}$ .

Raise $b$ to the power of $1$ .

$8 a^{9} (b^{1} b^{4})$

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

$8 a^{9} b^{1 + 4}$

Add $1$ and $4$ .

$8 a^{9} b^{5}$