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
Simplify. Assume that variableS can represent any real number.
![**Question:**
Write an equivalent expression using radical notation and simplify: \( 8^{2/3} \).
**Solution:**
To convert the expression \( 8^{2/3} \) into radical notation, we recognize that the exponent \(\frac{2}{3}\) can be rewritten to show both a root and a power.
The general form for converting fractional exponents \(a^{m/n}\) is:
\[a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\]
For \( 8^{2/3} \):
1. **Rewrite the base with the fractional exponent in radical form.**
\[ 8^{2/3} = \sqrt[3]{8^2} \]
2. **Simplify inside the radical first.**
\[8^2 = 64\]
\[ \sqrt[3]{8^2} = \sqrt[3]{64} \]
3. **Find the cube root of 64.**
\[64 = 4^3\]
\[ \sqrt[3]{64} = 4 \]
Therefore, the expression \(8^{2/3}\) in radical notation is \(\sqrt[3]{64}\) and the simplified result is \(4\).
\[ 8^{2/3} = \sqrt[3]{64} = 4 \]
This completes the solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94589190-d7d8-41be-9c6e-d2d51b756765%2F5bd5047b-08a2-47b7-9419-db9a20e89a69%2Fmgooo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Write an equivalent expression using radical notation and simplify: \( 8^{2/3} \).
**Solution:**
To convert the expression \( 8^{2/3} \) into radical notation, we recognize that the exponent \(\frac{2}{3}\) can be rewritten to show both a root and a power.
The general form for converting fractional exponents \(a^{m/n}\) is:
\[a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\]
For \( 8^{2/3} \):
1. **Rewrite the base with the fractional exponent in radical form.**
\[ 8^{2/3} = \sqrt[3]{8^2} \]
2. **Simplify inside the radical first.**
\[8^2 = 64\]
\[ \sqrt[3]{8^2} = \sqrt[3]{64} \]
3. **Find the cube root of 64.**
\[64 = 4^3\]
\[ \sqrt[3]{64} = 4 \]
Therefore, the expression \(8^{2/3}\) in radical notation is \(\sqrt[3]{64}\) and the simplified result is \(4\).
\[ 8^{2/3} = \sqrt[3]{64} = 4 \]
This completes the solution.
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