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
![**Mathematical Expression Simplification**
**Problem Statement:**
Given \( x > 0 \) and \( y > 0 \), select the expression that is equivalent to
\[ \sqrt[3]{-64x^{10}y^6} \]
**Solution Approach:**
Let's break down the expression inside the cube root:
1. First, recognize that \(-64\) can be factored and simplified as \(-64 = (-1) \times (4^3)\).
Thus, the expression inside the cube root becomes \[ \sqrt[3]{-1 \times 4^3 \times x^{10} \times y^6} \].
2. The cube root of a product is the product of the cube roots:
\[ \sqrt[3]{-1} \times \sqrt[3]{4^3} \times \sqrt[3]{x^{10}} \times \sqrt[3]{y^6} \]
3. Simplify each term individually:
- \(\sqrt[3]{-1} = -1\)
- \(\sqrt[3]{4^3} = 4\)
- For the variable \(x\):
\[
\sqrt[3]{x^{10}} = x^{\frac{10}{3}}
\]
- For the variable \(y\):
\[
\sqrt[3]{y^6} = y^2
\]
4. Combine the simplified terms:
\[ -1 \times 4 \times x^{\frac{10}{3}} \times y^2 = -4x^{\frac{10}{3}}y^2\]
**Equivalent Expression:**
\[ -4x^{\frac{10}{3}}y^2 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd778ba75-86ed-4d38-af28-a715226ea6d4%2F6ebb8257-416e-4170-a527-d098548dc50b%2Fhb4oar_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematical Expression Simplification**
**Problem Statement:**
Given \( x > 0 \) and \( y > 0 \), select the expression that is equivalent to
\[ \sqrt[3]{-64x^{10}y^6} \]
**Solution Approach:**
Let's break down the expression inside the cube root:
1. First, recognize that \(-64\) can be factored and simplified as \(-64 = (-1) \times (4^3)\).
Thus, the expression inside the cube root becomes \[ \sqrt[3]{-1 \times 4^3 \times x^{10} \times y^6} \].
2. The cube root of a product is the product of the cube roots:
\[ \sqrt[3]{-1} \times \sqrt[3]{4^3} \times \sqrt[3]{x^{10}} \times \sqrt[3]{y^6} \]
3. Simplify each term individually:
- \(\sqrt[3]{-1} = -1\)
- \(\sqrt[3]{4^3} = 4\)
- For the variable \(x\):
\[
\sqrt[3]{x^{10}} = x^{\frac{10}{3}}
\]
- For the variable \(y\):
\[
\sqrt[3]{y^6} = y^2
\]
4. Combine the simplified terms:
\[ -1 \times 4 \times x^{\frac{10}{3}} \times y^2 = -4x^{\frac{10}{3}}y^2\]
**Equivalent Expression:**
\[ -4x^{\frac{10}{3}}y^2 \]
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