Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 74
Simplify each complex fraction. $$ \frac{1}{\frac{b}{a}-\frac{a}{b}} $$
Short Answer
Expert verified
The simplified expression is \( \frac{ab}{(b-a)(b+a)} \).
Step by step solution
01
Identify the expression
The given complex fraction is \( \frac{1}{\frac{b}{a} - \frac{a}{b}} \). We need to simplify this expression.
02
Find a common denominator for the fractions in the denominator
The fractions in the denominator are \( \frac{b}{a} \) and \( \frac{a}{b} \). The common denominator for these fractions is \( ab \).
03
Rewrite each fraction with the common denominator
Rewrite \( \frac{b}{a} \) as \( \frac{b \cdot b}{a \cdot b} = \frac{b^2}{ab} \) and \( \frac{a}{b} \) as \( \frac{a \cdot a}{b \cdot a} = \frac{a^2}{ab} \).
04
Simplify the expression in the denominator
Now, subtract the fractions: \( \frac{b^2}{ab} - \frac{a^2}{ab} = \frac{b^2 - a^2}{ab} \).
05
Simplify \( \frac{1}{\frac{b^2 - a^2}{ab}} \)
The expression can be rewritten as \( \frac{ab}{b^2 - a^2} \), since dividing by a fraction is equivalent to multiplying by its reciprocal.
06
Factor the difference of squares in the denominator
The term \( b^2 - a^2 \) is a difference of squares, which can be factored as \( (b-a)(b+a) \).
07
Final simplified expression
Substitute the factored expression into \( \frac{ab}{b^2 - a^2} \) to get \( \frac{ab}{(b-a)(b+a)} \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Denominator
When simplifying complex fractions, finding a common denominator is a crucial first step. It helps unify different fractions under a single denominator, making it easier to perform arithmetic operations. In our exercise, we encounter two fractions within the denominator: \( \frac{b}{a} \) and \( \frac{a}{b} \). To find a common denominator between these, we look for a number or expression that both denominators, \( a \) and \( b \), can divide into evenly.
- The common denominator for \( \frac{b}{a} \) and \( \frac{a}{b} \) is the product of both denominators: \( ab \).
- Rewriting fractions using a common denominator allows us to manipulate them as if they were whole numbers.
Difference of Squares
The expression \( b^2 - a^2 \) is recognized as a difference of squares, a concept you will encounter frequently in algebra. A difference of squares can always be factored using the identity: \[(x^2 - y^2 = (x-y)(x+y)\] This identity simplifies algebraic expressions and is particularly helpful when trying to simplify complex fractions. In the given exercise:
- We identify \( b^2 - a^2 \) within our fraction.
- This expression can be factored as \( (b-a)(b+a) \).
Fraction Simplification
Simplifying fractions to their simplest form includes reducing complex expressions into single, manageable fractions. The given problem involves a complex fraction \( \frac{1}{\frac{b^2 - a^2}{ab}} \).
- The first simplification involves recognizing that dividing by a fraction is the same as multiplying by its reciprocal. Thus, \( \frac{1}{\frac{b^2 - a^2}{ab}} \) becomes \( \frac{ab}{b^2 - a^2} \).
- Use the factored form \( \frac{ab}{(b-a)(b+a)} \) for simplification.
