Simplify 0.2a^2b^3*(-5a^3b)^2

$0.2 a^{2} b^{3} \cdot {(- 5 a^{3} b)}^{2}$

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

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

$0.2 a^{2} b^{3} \cdot ({(- 5 a^{3})}^{2} b^{2})$

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

$0.2 a^{2} b^{3} \cdot ({(- 5)}^{2} {(a^{3})}^{2} b^{2})$

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

Move $b^{2}$ .

$0.2 a^{2} (b^{2} b^{3}) \cdot ({(- 5)}^{2} {(a^{3})}^{2})$

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

$0.2 a^{2} b^{2 + 3} \cdot ({(- 5)}^{2} {(a^{3})}^{2})$

Add $2$ and $3$ .

$0.2 a^{2} b^{5} \cdot ({(- 5)}^{2} {(a^{3})}^{2})$

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

$0.2 a^{2} b^{5} \cdot (25 {(a^{3})}^{2})$

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

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

$0.2 a^{2} b^{5} \cdot (25 a^{3 \cdot 2})$

Multiply $3$ by $2$ .

$0.2 a^{2} b^{5} \cdot (25 a^{6})$

Multiply $a^{2}$ by $a^{6}$ by adding the exponents.

Move $a^{6}$ .

$0.2 (a^{6} a^{2}) b^{5} \cdot 25$

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

$0.2 a^{6 + 2} b^{5} \cdot 25$

Add $6$ and $2$ .

$0.2 a^{8} b^{5} \cdot 25$

Multiply $25$ by $0.2$ .

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