Simplify (5ab^3)^2(-3a^4)

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

Rewrite using the commutative property of multiplication.

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

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

Apply the product rule to $5 a b^{3}$ .

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

Apply the product rule to $5 a$ .

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

Multiply $a^{2}$ by $a^{4}$ by adding the exponents.

Move $a^{4}$ .

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

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

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

Add $4$ and $2$ .

$- 3 (5^{2} a^{6} {(b^{3})}^{2})$

Raise $5$ to the power of $2$ .

$- 3 (25 a^{6} {(b^{3})}^{2})$

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

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

$- 3 (25 a^{6} b^{3 \cdot 2})$

Multiply $3$ by $2$ .

$- 3 (25 a^{6} b^{6})$

Multiply $25$ by $- 3$ .

$- 75 a^{6} b^{6}$