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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![## Algebraic Simplification
### Express in Simplest Radical Form
Given expression:
\[ \left(125x^6\right)^{\frac{5}{3}} \]
To solve this, express each component inside the parentheses in its simplest radical form.
First, break down the expression inside the parentheses:
\[ 125 = 5^3 \]
\[ x^6 = \left(x^2\right)^3 \]
So,
\[ \left(125x^6\right)^{\frac{5}{3}} = \left(5^3 \cdot \left(x^2\right)^3\right)^{\frac{5}{3}} \]
Apply the property of exponents \(\left(a \cdot b\right)^m = a^m \cdot b^m\):
\[ = \left(5^3\right)^{\frac{5}{3}} \cdot \left(\left(x^2\right)^3\right)^{\frac{5}{3}} \]
Simplify each term:
\[ \left(5^3\right)^{\frac{5}{3}} = 5^{3 \cdot \frac{5}{3}} = 5^5 \]
\[ \left(\left(x^2\right)^3\right)^{\frac{5}{3}} = \left(x^2\right)^{3 \cdot \frac{5}{3}} = x^{2 \cdot 5} = x^{10} \]
Combine the terms:
\[ = 5^5 \cdot x^{10} \]
Therefore, the simplest radical form is:
\[ 3125x^{10} \]
So we have:
\[ \left(125x^6\right)^{\frac{5}{3}} = 3125x^{10} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd778ba75-86ed-4d38-af28-a715226ea6d4%2F3654f133-9b70-4484-845b-257ab3e9dd32%2Ftghe9j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Algebraic Simplification
### Express in Simplest Radical Form
Given expression:
\[ \left(125x^6\right)^{\frac{5}{3}} \]
To solve this, express each component inside the parentheses in its simplest radical form.
First, break down the expression inside the parentheses:
\[ 125 = 5^3 \]
\[ x^6 = \left(x^2\right)^3 \]
So,
\[ \left(125x^6\right)^{\frac{5}{3}} = \left(5^3 \cdot \left(x^2\right)^3\right)^{\frac{5}{3}} \]
Apply the property of exponents \(\left(a \cdot b\right)^m = a^m \cdot b^m\):
\[ = \left(5^3\right)^{\frac{5}{3}} \cdot \left(\left(x^2\right)^3\right)^{\frac{5}{3}} \]
Simplify each term:
\[ \left(5^3\right)^{\frac{5}{3}} = 5^{3 \cdot \frac{5}{3}} = 5^5 \]
\[ \left(\left(x^2\right)^3\right)^{\frac{5}{3}} = \left(x^2\right)^{3 \cdot \frac{5}{3}} = x^{2 \cdot 5} = x^{10} \]
Combine the terms:
\[ = 5^5 \cdot x^{10} \]
Therefore, the simplest radical form is:
\[ 3125x^{10} \]
So we have:
\[ \left(125x^6\right)^{\frac{5}{3}} = 3125x^{10} \]
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