Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Topic: Simplifying Expressions Involving Exponents**
In the expression:
$$\frac{\sqrt{a} \sqrt[3]{b}}{\sqrt{a^5 b}},$$
when simplified, the exponent of \( a \) is [____] and the exponent of \( b \) is [____].
**Explanation:**
This problem involves simplifying an expression with roots and exponents. The given expression is a fraction with multiple root terms. To solve for the exponents of \( a \) and \( b \) once simplified, the rules of exponents and roots will be applied.
**Steps to Simplify:**
1. **Rewrite the Roots as Exponents:**
- \(\sqrt{a}\) can be written as \(a^{1/2}\).
- \(\sqrt[3]{b}\) can be written as \(b^{1/3}\).
- \(\sqrt{a^5 b}\) can be written as \((a^5 b)^{1/2} = a^{5/2} b^{1/2}\).
2. **Combine the Exponent Terms:**
- The numerator becomes \(a^{1/2} b^{1/3}\).
- The denominator is \(a^{5/2} b^{1/2}\).
3. **Subtract Exponents in the Denominator from the Numerator:**
- \(a\)'s exponent in the numerator is \(1/2\) and in the denominator is \(5/2\).
- \(1/2 - 5/2 = -2\).
- \(b\)'s exponent in the numerator is \(1/3\) and in the denominator is \(1/2\).
- \(1/3 - 1/2 = -1/6\).
4. **Final Simplified Exponents:**
- The exponent of \(a\) is \(-2\).
- The exponent of \(b\) is \(-1/6\).
Using this information, fill in the blanks in the provided format.
---
**Educational Objectives:**
- To strengthen understanding of exponent laws and transformations.
- To practice breaking down complex fraction expressions with roots.
- To reinforce simplification techniques involving exponents.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aaf72ae-8073-4cb4-ae75-7d87fdc3f506%2F2a16536f-6017-4024-b5e3-037ce88be4ab%2Fn0vdoq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Simplifying Expressions Involving Exponents**
In the expression:
$$\frac{\sqrt{a} \sqrt[3]{b}}{\sqrt{a^5 b}},$$
when simplified, the exponent of \( a \) is [____] and the exponent of \( b \) is [____].
**Explanation:**
This problem involves simplifying an expression with roots and exponents. The given expression is a fraction with multiple root terms. To solve for the exponents of \( a \) and \( b \) once simplified, the rules of exponents and roots will be applied.
**Steps to Simplify:**
1. **Rewrite the Roots as Exponents:**
- \(\sqrt{a}\) can be written as \(a^{1/2}\).
- \(\sqrt[3]{b}\) can be written as \(b^{1/3}\).
- \(\sqrt{a^5 b}\) can be written as \((a^5 b)^{1/2} = a^{5/2} b^{1/2}\).
2. **Combine the Exponent Terms:**
- The numerator becomes \(a^{1/2} b^{1/3}\).
- The denominator is \(a^{5/2} b^{1/2}\).
3. **Subtract Exponents in the Denominator from the Numerator:**
- \(a\)'s exponent in the numerator is \(1/2\) and in the denominator is \(5/2\).
- \(1/2 - 5/2 = -2\).
- \(b\)'s exponent in the numerator is \(1/3\) and in the denominator is \(1/2\).
- \(1/3 - 1/2 = -1/6\).
4. **Final Simplified Exponents:**
- The exponent of \(a\) is \(-2\).
- The exponent of \(b\) is \(-1/6\).
Using this information, fill in the blanks in the provided format.
---
**Educational Objectives:**
- To strengthen understanding of exponent laws and transformations.
- To practice breaking down complex fraction expressions with roots.
- To reinforce simplification techniques involving exponents.
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