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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Question
![**Educational Content: Simplifying Radicals with Rational Exponents**
**Problem Statement:**
Rewrite the following radical expression with a rational exponent:
\[
\sqrt[3]{\left(\frac{5x}{y}\right)^9}
\]
**Solution Prompt:**
Next to the expression, there's a box indicating where to input the simplified answer. Students are instructed to simplify their answer using integers or fractions for any numbers in the expression.
**Step-by-Step Explanation:**
1. **Identify the Radical Form:** The given expression is a cube root of a fraction raised to a power.
2. **Convert to Rational Exponent Form:**
- The cube root can be expressed as raising to the power of \(1/3\).
- Therefore, the expression \( \sqrt[3]{a} \) is equivalent to \( a^{1/3} \).
- Apply this to the entire fraction:
\[
\left(\frac{5x}{y}\right)^{9} \) raised to the \(1/3 \) power becomes \( \left(\frac{5x}{y}\right)^{9 \cdot \frac{1}{3}}
\]
3. **Simplify the Exponent:**
- Multiply the exponents: \( 9 \cdot \frac{1}{3} = 3 \).
- The expression simplifies to \( \left(\frac{5x}{y}\right)^3 \).
**Final Simplified Expression:**
\[
\left(\frac{5x}{y}\right)^3
\]
Students should complete the process by entering the final expression in the provided box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b4c38d1-e61f-4ae3-a015-72518f1ff016%2F9d083ab2-f911-4ab4-bad4-cdc09275b2e3%2Fkqxptvl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Simplifying Radicals with Rational Exponents**
**Problem Statement:**
Rewrite the following radical expression with a rational exponent:
\[
\sqrt[3]{\left(\frac{5x}{y}\right)^9}
\]
**Solution Prompt:**
Next to the expression, there's a box indicating where to input the simplified answer. Students are instructed to simplify their answer using integers or fractions for any numbers in the expression.
**Step-by-Step Explanation:**
1. **Identify the Radical Form:** The given expression is a cube root of a fraction raised to a power.
2. **Convert to Rational Exponent Form:**
- The cube root can be expressed as raising to the power of \(1/3\).
- Therefore, the expression \( \sqrt[3]{a} \) is equivalent to \( a^{1/3} \).
- Apply this to the entire fraction:
\[
\left(\frac{5x}{y}\right)^{9} \) raised to the \(1/3 \) power becomes \( \left(\frac{5x}{y}\right)^{9 \cdot \frac{1}{3}}
\]
3. **Simplify the Exponent:**
- Multiply the exponents: \( 9 \cdot \frac{1}{3} = 3 \).
- The expression simplifies to \( \left(\frac{5x}{y}\right)^3 \).
**Final Simplified Expression:**
\[
\left(\frac{5x}{y}\right)^3
\]
Students should complete the process by entering the final expression in the provided box.
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