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8. 81-1/4

8. 81-1/4

Algebra and Trigonometry (6th Edition)
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,...
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rewrite the number without radicals or exponents

**Problem Statement**: Simplify the expression \( 81^{-1/4} \). **Solution**: To simplify the expression \( 81^{-1/4} \), follow these steps: 1. **Identify the Base and Exponent:** - Base: 81 - Exponent: \(-1/4\) 2. **Understand the Meaning of the Exponent:** - The negative sign in the exponent indicates that you will take the reciprocal of the base. - The fraction \(-1/4\) means you take the fourth root of the reciprocal. 3. **Calculate the Reciprocal:** - The reciprocal of 81 is \( \frac{1}{81} \). 4. **Apply the Root:** - Find the fourth root of \(\frac{1}{81}\). - The fourth root of 81 is 3 (since \(3^4 = 81\)). 5. **Apply the Negative Exponent Rule:** - Thus, \( 81^{-1/4} = \left(\frac{1}{81}\right)^{1/4} = \frac{1}{3} \). **Conclusion**: The simplified form of \( 81^{-1/4} \) is \( \frac{1}{3} \). This demonstrates how to handle expressions with negative fractional exponents by taking the reciprocal and applying the root according to the exponent's fraction.
Transcribed Image Text:**Problem Statement**: Simplify the expression \( 81^{-1/4} \). **Solution**: To simplify the expression \( 81^{-1/4} \), follow these steps: 1. **Identify the Base and Exponent:** - Base: 81 - Exponent: \(-1/4\) 2. **Understand the Meaning of the Exponent:** - The negative sign in the exponent indicates that you will take the reciprocal of the base. - The fraction \(-1/4\) means you take the fourth root of the reciprocal. 3. **Calculate the Reciprocal:** - The reciprocal of 81 is \( \frac{1}{81} \). 4. **Apply the Root:** - Find the fourth root of \(\frac{1}{81}\). - The fourth root of 81 is 3 (since \(3^4 = 81\)). 5. **Apply the Negative Exponent Rule:** - Thus, \( 81^{-1/4} = \left(\frac{1}{81}\right)^{1/4} = \frac{1}{3} \). **Conclusion**: The simplified form of \( 81^{-1/4} \) is \( \frac{1}{3} \). This demonstrates how to handle expressions with negative fractional exponents by taking the reciprocal and applying the root according to the exponent's fraction.
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