Question
Question \( 1-10 \) Which expression is equivalent to \( 2.5^{2-3 y} \) ? \( \begin{array}{l}\frac{6.25}{(15.625)^{-y}} \\ 6.25(2.5)^{3 y} \\ 6.25-(2.5)^{3 y} \\ 6.25(15.625)^{-y}\end{array} \)
Ask by Clark Hill. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equivalent expression is \( 6.25(15.625)^{-y} \).
Solution
To determine which expression is equivalent to \( 2.5^{2 - 3y} \), let's simplify and compare each option.
1. **Original Expression:**
\[
2.5^{2 - 3y} = 2.5^2 \times 2.5^{-3y} = 6.25 \times 2.5^{-3y}
\]
2. **Option Analysis:**
- **Option 1:**
\[
\frac{6.25}{(15.625)^{-y}} = 6.25 \times (15.625)^y
\]
Since \( 15.625 = 2.5^3 \), this becomes:
\[
6.25 \times (2.5^3)^y = 6.25 \times 2.5^{3y}
\]
This does **not** match the original expression.
- **Option 2:**
\[
6.25 \times (2.5)^{3y}
\]
This is similar to Option 1 and also does **not** match the original expression.
- **Option 3:**
\[
6.25 - (2.5)^{3y}
\]
This subtractive form does **not** align with the multiplicative form of the original expression.
- **Option 4:**
\[
6.25 \times (15.625)^{-y}
\]
Simplifying \( 15.625 = 2.5^3 \):
\[
6.25 \times (2.5^3)^{-y} = 6.25 \times 2.5^{-3y}
\]
This matches the original expression perfectly:
\[
6.25 \times 2.5^{-3y} = 2.5^{2 - 3y}
\]
**Therefore, the correct equivalent expression is:**
**\( 6.25(15.625)^{-y} \)**
Answer: **6.25(15.625)^{-y}**
Reviewed and approved by the UpStudy tutoring team
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Explain 
Simplify this solution Mind Expander
To simplify \( 2.5^{2-3y} \), we can break it into two parts: \( 2.5^{2} \) and \( 2.5^{-3y} \). Since \( 2.5^{2} = 6.25 \) and \( 2.5^{-3y} = \frac{1}{2.5^{3y}} \), we can rewrite the expression as \( 6.25 \cdot 2.5^{-3y} \). Recognizing that \( 2.5^{-3y} \) can be expressed as \( \frac{1}{15.625^y} \) (since \( 15.625 = 2.5^4 \)) helps us express it in various forms. As a result, it leads us to identify that \( 6.25(15.625)^{-y} \) is equivalent to the original expression. So, the correct expression that is equivalent to \( 2.5^{2-3y} \) is: \( 6.25(15.625)^{-y} \).
