Elementary Algebra
17th Edition
ISBN:9780998625713
Author:Lynn Marecek, MaryAnne Anthony-Smith
Publisher:Lynn Marecek, MaryAnne Anthony-Smith
Chapter9: Roots And Radicals
Section9.4: Multiply Square Roots
Problem 313E: A garden will be made so as to contain two square sections, one section with side length 5+6 yards...
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![### Algebra Question: Equivalent Expressions
#### Question:
Select the expression that is equivalent to:
\[ \frac{1}{(3x^2 + 5)^{-\frac{5}{2}}} \]
#### Answer Choices:
1. \( \frac{1}{\sqrt{(3x^2 + 5)^5}} \)
2. \( \sqrt[5]{(3x^2 + 5)^2} \)
3. \( \frac{1}{\sqrt[5]{(3x^2 + 5)^2}} \)
4. \( \sqrt{(3x^2 + 5)^5} \)
#### Explanation:
To find the expression equivalent to \( \frac{1}{(3x^2 + 5)^{-\frac{5}{2}}} \), let's simplify it step by step:
1. Start with the given expression:
\[ \frac{1}{(3x^2 + 5)^{-\frac{5}{2}}} \]
2. Recognize that a negative exponent indicates a reciprocal:
\[ \frac{1}{(a^{-\frac{5}{2}})} = (a^{\frac{5}{2}}) \]
3. Apply this property:
\[ (3x^2 + 5)^{\frac{5}{2}} \]
4. Identify that the fractional exponent \( \frac{5}{2} \) can be broken into a square root and a power:
\[ (a^{\frac{5}{2}}) = (a^5)^{\frac{1}{2}} = \sqrt{(a^5)} \]
5. Apply this to the expression:
\[ \sqrt{(3x^2 + 5)^5} \]
Therefore, the correct answer is:
- \( \sqrt{(3x^2 + 5)^5} \)
#### Correct Choice:
4. \( \sqrt{(3x^2 + 5)^5} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd778ba75-86ed-4d38-af28-a715226ea6d4%2F7e0ebe95-29a3-41b2-9fd5-adf067fca021%2Fojr2ptg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Algebra Question: Equivalent Expressions
#### Question:
Select the expression that is equivalent to:
\[ \frac{1}{(3x^2 + 5)^{-\frac{5}{2}}} \]
#### Answer Choices:
1. \( \frac{1}{\sqrt{(3x^2 + 5)^5}} \)
2. \( \sqrt[5]{(3x^2 + 5)^2} \)
3. \( \frac{1}{\sqrt[5]{(3x^2 + 5)^2}} \)
4. \( \sqrt{(3x^2 + 5)^5} \)
#### Explanation:
To find the expression equivalent to \( \frac{1}{(3x^2 + 5)^{-\frac{5}{2}}} \), let's simplify it step by step:
1. Start with the given expression:
\[ \frac{1}{(3x^2 + 5)^{-\frac{5}{2}}} \]
2. Recognize that a negative exponent indicates a reciprocal:
\[ \frac{1}{(a^{-\frac{5}{2}})} = (a^{\frac{5}{2}}) \]
3. Apply this property:
\[ (3x^2 + 5)^{\frac{5}{2}} \]
4. Identify that the fractional exponent \( \frac{5}{2} \) can be broken into a square root and a power:
\[ (a^{\frac{5}{2}}) = (a^5)^{\frac{1}{2}} = \sqrt{(a^5)} \]
5. Apply this to the expression:
\[ \sqrt{(3x^2 + 5)^5} \]
Therefore, the correct answer is:
- \( \sqrt{(3x^2 + 5)^5} \)
#### Correct Choice:
4. \( \sqrt{(3x^2 + 5)^5} \)
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