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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![Below is the transcription and explanation of the content that will appear on the educational website:
---
### Understanding Radical and Exponential Expressions
**Problem:**
Choose the equivalent expression from the options below:
#### Option A:
\[
\frac{1}{\sqrt{(3x^2+5)^5}}
\]
#### Option B:
\[
\sqrt[5]{(3x^2+5)^2}
\]
#### Option C:
\[
\frac{1}{\sqrt[5]{(3x^2+5)^2}}
\]
#### Option D:
\[
\sqrt{(3x^2+5)^5}
\]
**Explanation of Each Option:**
- **Option A:** This expression represents the reciprocal of the square root of \((3x^2+5)^5\). The expression inside the square root is raised to the 5th power, and then the entire expression is taken as the reciprocal.
- **Option B:** This is the 5th root (\(\sqrt[5]{}\)) of the entire expression \((3x^2+5)\) raised to the power of 2.
- **Option C:** This represents the reciprocal of the 5th root of the expression \((3x^2+5)\) squared.
- **Option D:** This is the square root of the expression \((3x^2+5)\) raised to the 5th power.
Understanding how to manipulate and simplify radical and exponential expressions is fundamental in solving algebraic equations. Look closely at each option and apply the rules of exponents and roots to determine the correct equivalent expression.
---
Ensure you practice on similar problems to solidify your understanding of the concepts outlined.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd778ba75-86ed-4d38-af28-a715226ea6d4%2Fe76e5d10-54a8-4a10-b6d4-74dc4b5dbcd9%2Fmp0g10b_processed.png&w=3840&q=75)
Transcribed Image Text:Below is the transcription and explanation of the content that will appear on the educational website:
---
### Understanding Radical and Exponential Expressions
**Problem:**
Choose the equivalent expression from the options below:
#### Option A:
\[
\frac{1}{\sqrt{(3x^2+5)^5}}
\]
#### Option B:
\[
\sqrt[5]{(3x^2+5)^2}
\]
#### Option C:
\[
\frac{1}{\sqrt[5]{(3x^2+5)^2}}
\]
#### Option D:
\[
\sqrt{(3x^2+5)^5}
\]
**Explanation of Each Option:**
- **Option A:** This expression represents the reciprocal of the square root of \((3x^2+5)^5\). The expression inside the square root is raised to the 5th power, and then the entire expression is taken as the reciprocal.
- **Option B:** This is the 5th root (\(\sqrt[5]{}\)) of the entire expression \((3x^2+5)\) raised to the power of 2.
- **Option C:** This represents the reciprocal of the 5th root of the expression \((3x^2+5)\) squared.
- **Option D:** This is the square root of the expression \((3x^2+5)\) raised to the 5th power.
Understanding how to manipulate and simplify radical and exponential expressions is fundamental in solving algebraic equations. Look closely at each option and apply the rules of exponents and roots to determine the correct equivalent expression.
---
Ensure you practice on similar problems to solidify your understanding of the concepts outlined.
![**Objective:** Express the following expression in its simplest radical form.
**Problem:**
\[ (125x^6)^{\frac{5}{3}} \]
**Explanation:**
To simplify the expression \((125x^6)^{\frac{5}{3}}\), we need to follow the steps below:
1. **Break down the base terms:**
- \(125\) and \(x^6\) are the base terms.
2. **Express the number 125 as a power of a number:**
- Notice that \(125 = 5^3\).
3. **Substitute in the expression:**
- Replace \(125\) with \(5^3\).
This changes the expression to:
\[ ((5^3)x^6)^{\frac{5}{3}} \]
4. **Distribute the exponent \(\frac{5}{3}\) to both factors inside the parenthesis:**
- Apply the power rule of exponents: \((a \cdot b)^c = a^c \cdot b^c\).
Therefore:
\[ (5^3)^{\frac{5}{3}} \cdot (x^6)^{\frac{5}{3}} \]
5. **Simplify each part:**
- Recall the power rule \((a^m)^n = a^{m \cdot n}\):
- \((5^3)^{\frac{5}{3}} = 5^{3 \cdot \frac{5}{3}} = 5^5\).
- \((x^6)^{\frac{5}{3}} = x^{6 \cdot \frac{5}{3}} = x^{10}\).
Finally, the expression becomes:
\[ 5^5 \cdot x^{10} \]
**Conclusion:**
\[ (125x^6)^{\frac{5}{3}} = 5^5 \cdot x^{10} \]
So, the expression in its simplest radical form is \(5^5 x^{10}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd778ba75-86ed-4d38-af28-a715226ea6d4%2Fe76e5d10-54a8-4a10-b6d4-74dc4b5dbcd9%2F6rhnkgp_processed.png&w=3840&q=75)
Transcribed Image Text:**Objective:** Express the following expression in its simplest radical form.
**Problem:**
\[ (125x^6)^{\frac{5}{3}} \]
**Explanation:**
To simplify the expression \((125x^6)^{\frac{5}{3}}\), we need to follow the steps below:
1. **Break down the base terms:**
- \(125\) and \(x^6\) are the base terms.
2. **Express the number 125 as a power of a number:**
- Notice that \(125 = 5^3\).
3. **Substitute in the expression:**
- Replace \(125\) with \(5^3\).
This changes the expression to:
\[ ((5^3)x^6)^{\frac{5}{3}} \]
4. **Distribute the exponent \(\frac{5}{3}\) to both factors inside the parenthesis:**
- Apply the power rule of exponents: \((a \cdot b)^c = a^c \cdot b^c\).
Therefore:
\[ (5^3)^{\frac{5}{3}} \cdot (x^6)^{\frac{5}{3}} \]
5. **Simplify each part:**
- Recall the power rule \((a^m)^n = a^{m \cdot n}\):
- \((5^3)^{\frac{5}{3}} = 5^{3 \cdot \frac{5}{3}} = 5^5\).
- \((x^6)^{\frac{5}{3}} = x^{6 \cdot \frac{5}{3}} = x^{10}\).
Finally, the expression becomes:
\[ 5^5 \cdot x^{10} \]
**Conclusion:**
\[ (125x^6)^{\frac{5}{3}} = 5^5 \cdot x^{10} \]
So, the expression in its simplest radical form is \(5^5 x^{10}\).
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