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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![### Algebra and Simplification: Understanding Equivalent Expressions
#### Problem Statement:
Which expression is equivalent to \(\sqrt{\frac{27x^3}{216y^6}}\)?
#### Options:
A. \(\frac{x}{2y^2}\)
B. \(\frac{x}{2y^3}\)
C. \(\frac{x}{8y^3}\)
D. \(\frac{x}{8y^2}\)
#### Explanation:
To solve this problem, we need to simplify the given square root expression. Let's break down the steps involved:
1. **Simplify the Fraction**:
\[\frac{27x^3}{216y^6}\]
First, simplify the numerical part:
\[\frac{27}{216} = \frac{1}{8}\]
After simplifying the numerical coefficients, this fraction becomes:
\[\frac{x^3}{8y^6}\]
2. **Apply Square Root**:
We need to apply the square root to both the numerator and the denominator:
\[\sqrt{\frac{x^3}{8y^6}} = \frac{\sqrt{x^3}}{\sqrt{8y^6}}\]
3. **Simplify the Square Roots**:
- For the numerator: \(\sqrt{x^3}\).
- For the denominator: \(\sqrt{8y^6}\).
Using the properties of exponents and square roots:
\[\sqrt{x^3} = x^{3/2}\]
\[\sqrt{8y^6} = \sqrt{8} \cdot \sqrt{y^6}\]
Further simplifying:
\[\sqrt{8} = 2\sqrt{2}\]
And,
\[\sqrt{y^6} = y^3\]
So the denominator becomes:
\[2\sqrt{2} \cdot y^3\]
4. **Combine and Simplify**:
\[\frac{x^{3/2}}{2y^3\sqrt{2}}\]
We note that there isn't an exact match to this format among the provided options, but considering the simplification of all parts, the term removed from the options is \(\sqrt{2}\) suggesting the simplified ratio without \(\sqrt{2}\) gives us close to one of the simplified](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffda00417-b7b2-4353-8e1a-cc43c4365f92%2F1b3ef461-f9e3-4275-9152-dc4de845127f%2Fi24s5cn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Algebra and Simplification: Understanding Equivalent Expressions
#### Problem Statement:
Which expression is equivalent to \(\sqrt{\frac{27x^3}{216y^6}}\)?
#### Options:
A. \(\frac{x}{2y^2}\)
B. \(\frac{x}{2y^3}\)
C. \(\frac{x}{8y^3}\)
D. \(\frac{x}{8y^2}\)
#### Explanation:
To solve this problem, we need to simplify the given square root expression. Let's break down the steps involved:
1. **Simplify the Fraction**:
\[\frac{27x^3}{216y^6}\]
First, simplify the numerical part:
\[\frac{27}{216} = \frac{1}{8}\]
After simplifying the numerical coefficients, this fraction becomes:
\[\frac{x^3}{8y^6}\]
2. **Apply Square Root**:
We need to apply the square root to both the numerator and the denominator:
\[\sqrt{\frac{x^3}{8y^6}} = \frac{\sqrt{x^3}}{\sqrt{8y^6}}\]
3. **Simplify the Square Roots**:
- For the numerator: \(\sqrt{x^3}\).
- For the denominator: \(\sqrt{8y^6}\).
Using the properties of exponents and square roots:
\[\sqrt{x^3} = x^{3/2}\]
\[\sqrt{8y^6} = \sqrt{8} \cdot \sqrt{y^6}\]
Further simplifying:
\[\sqrt{8} = 2\sqrt{2}\]
And,
\[\sqrt{y^6} = y^3\]
So the denominator becomes:
\[2\sqrt{2} \cdot y^3\]
4. **Combine and Simplify**:
\[\frac{x^{3/2}}{2y^3\sqrt{2}}\]
We note that there isn't an exact match to this format among the provided options, but considering the simplification of all parts, the term removed from the options is \(\sqrt{2}\) suggesting the simplified ratio without \(\sqrt{2}\) gives us close to one of the simplified
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