To determine which expression is equivalent to the given expression, [tex]\( \left(-3 x^3 y^2\right)\left(5 x^5 y^4\right)^2 \)[/tex], we will simplify the expression step-by-step.
1. Simplify the inner expression:
[tex]\[
\left(5 x^5 y^4\right)^2
\][/tex]
When we raise a product to a power, we raise each factor to that power:
[tex]\[
(5 x^5 y^4)^2 = 5^2 \cdot (x^5)^2 \cdot (y^4)^2
\][/tex]
Which simplifies to:
[tex]\[
= 25 \cdot x^{10} \cdot y^8
\][/tex]
2. Multiply by the outer expression:
[tex]\[
\left(-3 x^3 y^2\right) \cdot (25 x^{10} y^8)
\][/tex]
Apply the distributive property:
[tex]\[
-3 \cdot 25 \cdot x^3 \cdot x^{10} \cdot y^2 \cdot y^8
\][/tex]
Simplify the coefficients:
[tex]\[
-3 \times 25 = -75
\][/tex]
Apply the properties of exponents:
[tex]\[
x^3 \cdot x^{10} = x^{3+10} = x^{13}
\][/tex]
[tex]\[
y^2 \cdot y^8 = y^{2+8} = y^{10}
\][/tex]
Putting it all together, we obtain:
[tex]\[
-75 x^{13} y^{10}
\][/tex]
Thus, the expression that is equivalent to [tex]\( \left(-3 x^3 y^2\right)\left(5 x^5 y^4\right)^2 \)[/tex] is:
[tex]\[
-75 x^{13} y^{10}
\][/tex]
So, the correct answer is:
[tex]\[
-75 x^{13} y^{10}
\][/tex]
