To determine which expression is equivalent to [tex]\(\frac{5 y^3}{(5 y)^{-2}}\)[/tex], let's simplify it step-by-step.
1. Start with the given expression:
[tex]\[
\frac{5 y^3}{(5 y)^{-2}}
\][/tex]
2. Consider the denominator [tex]\((5 y)^{-2}\)[/tex]. When raising a product to a power, distribute the exponent to each factor in the product:
[tex]\[
(5 y)^{-2} = 5^{-2} y^{-2}
\][/tex]
3. Now substitute [tex]\(5^{-2} y^{-2}\)[/tex] back into the original expression:
[tex]\[
\frac{5 y^3}{5^{-2} y^{-2}}
\][/tex]
4. Knowing that dividing by a fraction is the same as multiplying by its reciprocal, we rewrite the expression:
[tex]\[
\frac{5 y^3}{5^{-2} y^{-2}} = 5 y^3 \cdot 5^2 y^2
\][/tex]
5. Multiply the coefficients and combine the exponents for [tex]\(y\)[/tex]:
[tex]\[
5 y^3 \cdot 5^2 y^2 = (5 \cdot 5^2) \cdot (y^3 \cdot y^2)
\][/tex]
Simplify each part:
[tex]\[
5 \cdot 5^2 = 5 \cdot 25 = 125
\][/tex]
[tex]\[
y^3 \cdot y^2 = y^{3+2} = y^5
\][/tex]
6. Combine these simplified parts to get the final simplified expression:
[tex]\[
125 y^5
\][/tex]
Therefore, the expression [tex]\(\frac{5 y^3}{(5 y)^{-2}}\)[/tex] simplifies to:
[tex]\[
125 y^5
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{125 y^5}
\][/tex]
