To determine which expression is equivalent to [tex]\(\frac{5 y^3}{(5 y)^{-2}}\)[/tex], let's simplify the given expression step by step:
1. Original expression:
[tex]\[
\frac{5y^3}{(5y)^{-2}}
\][/tex]
2. Simplify the denominator [tex]\((5y)^{-2}\)[/tex]:
The expression [tex]\((5y)^{-2}\)[/tex] can be rewritten using the property of exponents which states that [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]:
[tex]\[
(5y)^{-2} = \frac{1}{(5y)^2}
\][/tex]
3. Calculate [tex]\((5y)^2\)[/tex]:
To square the term [tex]\(5y\)[/tex]:
[tex]\[
(5y)^2 = 5^2 \cdot y^2 = 25y^2
\][/tex]
Therefore:
[tex]\[
(5y)^{-2} = \frac{1}{25y^2}
\][/tex]
4. Substitute the simplified denominator back into the original expression:
[tex]\[
\frac{5y^3}{\frac{1}{25y^2}}
\][/tex]
5. Simplify the complex fraction:
To simplify [tex]\(\frac{5y^3}{\frac{1}{25y^2}}\)[/tex], you can multiply the numerator by the reciprocal of the denominator:
[tex]\[
\frac{5y^3}{\frac{1}{25y^2}} = 5y^3 \cdot 25y^2
\][/tex]
6. Multiply the terms:
To multiply [tex]\(5y^3\)[/tex] by [tex]\(25y^2\)[/tex]:
[tex]\[
5 \cdot 25 \cdot y^3 \cdot y^2
\][/tex]
First, multiply the coefficients:
[tex]\[
5 \cdot 25 = 125
\][/tex]
Then, use the property of exponents which states that when you multiply like bases, you add the exponents:
[tex]\[
y^3 \cdot y^2 = y^{3+2} = y^5
\][/tex]
7. Combine the results:
Therefore:
[tex]\[
5 \cdot 25 \cdot y^3 \cdot y^2 = 125y^5
\][/tex]
Hence, the expression [tex]\(\frac{5y^3}{(5y)^{-2}}\)[/tex] simplifies to:
[tex]\[
125 y^5
\][/tex]
So, the correct equivalent expression is:
[tex]\[
\boxed{125 y^5}
\][/tex]
