Question
Simplify. Write your answers without exponents. \( \left(\frac{1}{32}\right)^{\frac{3}{5}}=\square \)
Ask by Chandler Chambers. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\frac{1}{8}
\]
Solution
To simplify \( \left(\frac{1}{32}\right)^{\frac{3}{5}} \) without using exponents, follow these steps:
1. **Express 32 as a Power of 2:**
\[
32 = 2^5
\]
Therefore,
\[
\frac{1}{32} = 2^{-5}
\]
2. **Apply the Exponent:**
\[
\left(2^{-5}\right)^{\frac{3}{5}} = 2^{-3} = \frac{1}{2^3}
\]
3. **Calculate the Final Value:**
\[
\frac{1}{2^3} = \frac{1}{8}
\]
**Final Answer:**
\[
\frac{1}{8}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify \( \left(\frac{1}{32}\right)^{\frac{3}{5}} \), we can first express \( \frac{1}{32} \) as \( 32^{-1} \). This gives us \( (32^{-1})^{\frac{3}{5}} \), which simplifies to \( 32^{-\frac{3}{5}} \). Next, since \( 32 = 2^5 \), we can rewrite our expression as \( (2^5)^{-\frac{3}{5}} \). By multiplying the exponents, we get \( 2^{-3} \), which simplifies to \( \frac{1}{2^3} = \frac{1}{8} \). So, \( \left(\frac{1}{32}\right)^{\frac{3}{5}} = \frac{1}{8} \).
