Problem 29 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{a x+a b}{x^{2}-b^{2}}}{\frac{x+b}{x-b}} $$

Step-by-Step Solution

Answer

The simplified expression is \(a\).

Step 1: Understand the Problem

We have the complex fraction: \( \frac{\frac{a x+a b}{x^{2}-b^{2}}}{\frac{x+b}{x-b}} \). Our goal is to simplify it by transforming it into a simpler expression.

Step 2: Simplify the Denominator

The denominator is \( \frac{x+b}{x-b} \). A fraction divided by another fraction is the same as multiplying by the reciprocal of the divisor.

Step 3: Apply the Rule of Division

Rewrite the problem as a multiplication: \( \frac{a x+a b}{x^{2}-b^{2}} \times \frac{x-b}{x+b} \).

Step 4: Simplify the Numerator

The numerator can be written as \((a)(x+b)\), as \( ax + ab = a(x + b) \).

Step 5: Recognize the Difference of Squares in the Denominator

The expression \(x^2 - b^2\) is a difference of squares and can be factored as \((x-b)(x+b)\).

Step 6: Simplify the Fraction

The expression becomes \(\frac{a(x+b)}{(x-b)(x+b)} \times \frac{x-b}{x+b}\). Cancel out common factors \((x+b)\) and \((x-b)\), leading to \(a\cdot1\).

Step 7: Write the Final Simplified Expression

After simplifying, the final expression is \(a\).

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, we're essentially looking to reduce fractions to their simplest form. A fraction is simplified when the numerator and the denominator have no common factors other than 1. In mathematics, simplifying complex fractions is a key skill. It's important because it makes calculations easier and helps us better understand relationships in math problems.

• First, find any common factors that the numerator and the denominator share.
• Next, divide both numerator and denominator by their greatest common factor (GCF).

In the original exercise, simplifying involves rewriting a complex expression in a less complex form. By dealing with individual fractions in the numerator and the denominator separately, and then combining them according to fraction division rules, the expression becomes more manageable.

Difference of Squares

The difference of squares is a specific concept where we deal with the subtraction of two square numbers. Factoring such expressions is straightforward once we recognize the pattern. Consider an expression of the form \(a^2 - b^2\). This can be factored into \((a-b)(a+b)\).Recognizing the difference of squares is crucial in simplifying complex fractions, as seen in our original problem.

• Look for expressions in the form of \(x^2 - b^2\).
• Factor it using the identity \((x-b)(x+b)\).

This pattern is not only a powerful tool in algebra, but also simplifies many seemingly difficult problems. In our exercise, identifying \(x^2-b^2\) as a difference of squares allowed us to factor and simplify effectively.

Factoring Expressions

Factoring is the process of breaking an expression down into a product of simpler expressions, or "factors". It's a fundamental aspect of algebra and is extremely useful when simplifying fractions and solving equations. To factor expressions like \(ax + ab\), we look for common terms that can be grouped.

• Identify common terms in each part of the expression.
• Use these terms to express the overall expression as a product of factors.

In our original problem, \(ax + ab\) is factored as \(a(x+b)\). This lets us simplify by canceling out terms more easily in the complex fraction. Recognizing and applying factoring allows us to transform complex problems into simpler tasks.

Reciprocal Multiplication

Reciprocal multiplication is a technique used to divide one fraction by another. When we divide by a fraction, we multiply by its reciprocal. This operation is especially useful for tackling complex fractions, as showcased in the exercise.

Identify the reciprocal of the divisor fraction.

Multiply the dividend by this reciprocal.

In our problem, we divided \(\frac{ax + ab}{x^2 - b^2} \) by \(\frac{x+b}{x-b}\), which is equivalent to multiplying by the reciprocal \(\frac{x-b}{x+b}\). This technique incredibly simplifies the fraction division, transforming it into straightforward multiplication and making it easier to manage and simplify further.