Chapter8: Roots And Radicals
Section8.5: Divide Radical Expressions
Problem 284E: Explain what is meant by the word rationalize in the phrase, "rationalize a denominator."
Related questions
Question
![**Simplify the Given Expression:**
\[ \frac{a + 2}{a^2 + a - 2} + \frac{4}{a^2 + 2a - 3} \]
### Steps to Simplify:
1. **Factor the Denominators:**
- The denominator \(a^2 + a - 2\) can be factored as:
\[ a^2 + a - 2 = (a + 2)(a - 1) \]
- The denominator \(a^2 + 2a - 3\) can be factored as:
\[ a^2 + 2a - 3 = (a + 3)(a - 1) \]
2. **Rewrite the Original Expression with Factored Denominators:**
\[ \frac{a + 2}{(a + 2)(a - 1)} + \frac{4}{(a + 3)(a - 1)} \]
3. **Identify the Common Denominator:**
The common denominator for this expression is \((a+2)(a-1)(a+3)\).
4. **Rewrite Each Fraction with the Common Denominator:**
- For the first fraction:
\[ \frac{a + 2}{(a + 2)(a - 1)} = \frac{a + 2}{(a + 2)(a - 1)} \times \frac{a + 3}{a + 3} = \frac{(a+2)(a+3)}{(a + 2)(a - 1)(a + 3)} = \frac{a^2 + 5a + 6}{(a + 2)(a - 1)(a + 3)} \]
- For the second fraction:
\[ \frac{4}{(a + 3)(a - 1)} = \frac{4}{(a + 3)(a - 1)} \times \frac{a + 2}{a + 2} = \frac{4(a + 2)}{(a + 2)(a - 1)(a + 3)} = \frac{4a + 8}{(a + 2)(a - 1)(a + 3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d158991-3671-4459-aef6-bd0615668acc%2F621529bd-7eb6-4c87-8cf2-dcc6e36c341a%2Fhph961c_processed.png&w=3840&q=75)
Transcribed Image Text:**Simplify the Given Expression:**
\[ \frac{a + 2}{a^2 + a - 2} + \frac{4}{a^2 + 2a - 3} \]
### Steps to Simplify:
1. **Factor the Denominators:**
- The denominator \(a^2 + a - 2\) can be factored as:
\[ a^2 + a - 2 = (a + 2)(a - 1) \]
- The denominator \(a^2 + 2a - 3\) can be factored as:
\[ a^2 + 2a - 3 = (a + 3)(a - 1) \]
2. **Rewrite the Original Expression with Factored Denominators:**
\[ \frac{a + 2}{(a + 2)(a - 1)} + \frac{4}{(a + 3)(a - 1)} \]
3. **Identify the Common Denominator:**
The common denominator for this expression is \((a+2)(a-1)(a+3)\).
4. **Rewrite Each Fraction with the Common Denominator:**
- For the first fraction:
\[ \frac{a + 2}{(a + 2)(a - 1)} = \frac{a + 2}{(a + 2)(a - 1)} \times \frac{a + 3}{a + 3} = \frac{(a+2)(a+3)}{(a + 2)(a - 1)(a + 3)} = \frac{a^2 + 5a + 6}{(a + 2)(a - 1)(a + 3)} \]
- For the second fraction:
\[ \frac{4}{(a + 3)(a - 1)} = \frac{4}{(a + 3)(a - 1)} \times \frac{a + 2}{a + 2} = \frac{4(a + 2)}{(a + 2)(a - 1)(a + 3)} = \frac{4a + 8}{(a + 2)(a - 1)(a + 3
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 4 images
Recommended textbooks for you
Elementary Algebra
Algebra
ISBN:
9780998625713
Author:
Lynn Marecek, MaryAnne Anthony-Smith
Publisher:
OpenStax - Rice University
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Elementary Algebra
Algebra
ISBN:
9780998625713
Author:
Lynn Marecek, MaryAnne Anthony-Smith
Publisher:
OpenStax - Rice University
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell