Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![**Problem Statement:**
Write \(32^{-\frac{2}{5}}\) without using exponents or radicals.
**Explanation:**
The expression involves a fractional exponent, where 32 is raised to the power of \(-\frac{2}{5}\). To simplify this expression without exponents or radicals, follow these steps:
1. **Understand the Negative Exponent:**
A negative exponent indicates taking the reciprocal of the base raised to the corresponding positive exponent. Thus, \(32^{-\frac{2}{5}} = \frac{1}{32^{\frac{2}{5}}}\).
2. **Simplify the Fractional Exponent:**
The exponent \(\frac{2}{5}\) suggests taking the fifth root of 32 first and then squaring the result. The fifth root of 32 is the number which, when raised to the power 5, equals 32. Since \(2^5 = 32\), the fifth root of 32 is 2.
3. **Apply the Positive Exponent:**
After finding the fifth root, which is 2, you then square it to resolve the exponent of 2. Thus:
\[
32^{\frac{2}{5}} = (32^{\frac{1}{5}})^2 = 2^2 = 4
\]
4. **Reciprocal of the Squared Number:**
Finally, since we have a negative exponent originally, take the reciprocal:
\[
32^{-\frac{2}{5}} = \frac{1}{4}
\]
**Solution:**
Therefore, \(32^{-\frac{2}{5}} = \frac{1}{4}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F407473a0-45d3-49b9-96cd-c9be6629504f%2F48e6509c-f3d7-4aa9-9e51-1eb318884fd9%2Fc9zjsu_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Write \(32^{-\frac{2}{5}}\) without using exponents or radicals.
**Explanation:**
The expression involves a fractional exponent, where 32 is raised to the power of \(-\frac{2}{5}\). To simplify this expression without exponents or radicals, follow these steps:
1. **Understand the Negative Exponent:**
A negative exponent indicates taking the reciprocal of the base raised to the corresponding positive exponent. Thus, \(32^{-\frac{2}{5}} = \frac{1}{32^{\frac{2}{5}}}\).
2. **Simplify the Fractional Exponent:**
The exponent \(\frac{2}{5}\) suggests taking the fifth root of 32 first and then squaring the result. The fifth root of 32 is the number which, when raised to the power 5, equals 32. Since \(2^5 = 32\), the fifth root of 32 is 2.
3. **Apply the Positive Exponent:**
After finding the fifth root, which is 2, you then square it to resolve the exponent of 2. Thus:
\[
32^{\frac{2}{5}} = (32^{\frac{1}{5}})^2 = 2^2 = 4
\]
4. **Reciprocal of the Squared Number:**
Finally, since we have a negative exponent originally, take the reciprocal:
\[
32^{-\frac{2}{5}} = \frac{1}{4}
\]
**Solution:**
Therefore, \(32^{-\frac{2}{5}} = \frac{1}{4}\).
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