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

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

Step 1

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

${(- 1)}^{4} {(a^{2})}^{4} \cdot {(3 a b^{5})}^{3}$

Step 2

Raise $- 1$ to the power of $4$ .

$1 {(a^{2})}^{4} \cdot {(3 a b^{5})}^{3}$

Step 3

Multiply ${(a^{2})}^{4}$ by $1$ .

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

Step 4

Multiply the exponents in ${(a^{2})}^{4}$ .

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

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

Multiply $2$ by $4$ .

$a^{8} \cdot {(3 a b^{5})}^{3}$

Step 5

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

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

$a^{8} \cdot ({(3 a)}^{3} {(b^{5})}^{3})$

Apply the product rule to $3 a$ .

$a^{8} \cdot (3^{3} a^{3} {(b^{5})}^{3})$

Step 6

Rewrite using the commutative property of multiplication.

$3^{3} a^{8} (a^{3} {(b^{5})}^{3})$

Step 7

Multiply $a^{8}$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

$3^{3} (a^{3} a^{8}) {(b^{5})}^{3}$

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

$3^{3} a^{3 + 8} {(b^{5})}^{3}$

Add $3$ and $8$ .

$3^{3} a^{11} {(b^{5})}^{3}$

Step 8

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

$27 a^{11} {(b^{5})}^{3}$

Step 9

Multiply the exponents in ${(b^{5})}^{3}$ .

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

$27 a^{11} b^{5 \cdot 3}$

Multiply $5$ by $3$ .

$27 a^{11} b^{15}$