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------------------------------------------------ Which expression is equivalent to [tex]$2.5^{2-3y}$[/tex]?

A. [tex]\frac{6.25}{(15.625)^{-y}}[/tex]
B. [tex]6.25(2.5)^{3y}[/tex]
C. [tex]6.25-(2.5)^{3y}[/tex]
D. [tex]6.25(15.625)^{-y}[/tex]

Answer :

To find an expression equivalent to [tex]\(2.5^{2-3y}\)[/tex], we can simplify the expression step by step. Here's how:

1. Understand the Original Expression:
The original expression is [tex]\(2.5^{2 - 3y}\)[/tex].

2. Break it Down:
Notice that the expression can be rewritten using exponent rules. Let's break it down as follows:
- [tex]\(2.5^{2 - 3y} = 2.5^2 \times 2.5^{-3y}\)[/tex]

3. Calculate [tex]\(2.5^2\)[/tex]:
First, calculate [tex]\(2.5^2\)[/tex]:
- [tex]\(2.5^2 = 6.25\)[/tex]

4. Re-express the Original Expression:
Substituting back, we get:
- [tex]\(2.5^{2 - 3y} = 6.25 \times 2.5^{-3y}\)[/tex]

5. Identify the Correct Answer:
By comparing with the given choices:
- The expression becomes [tex]\(6.25 \times (2.5)^{-3y}\)[/tex].

Now, recognize that [tex]\(2.5^{-3y}\)[/tex] can be rewritten in terms of another base. Since [tex]\((2.5)^3 = 15.625\)[/tex], then:
- [tex]\(2.5^{-3y} = (15.625)^{-y}\)[/tex]

6. Correct Choice:
Therefore, the expression simplifies to:
- [tex]\(6.25 \times (15.625)^{-y}\)[/tex]

This matches the choice:
- [tex]\(6.25(15.625)^{-y}\)[/tex]

So, the equivalent expression is [tex]\(6.25(15.625)^{-y}\)[/tex].