Problem 53 - Algebra for College Students

Question

Simplify. $$\frac{a^{-1}-b^{-1}}{a-b}$$

Step-by-Step Solution

Answer

\(\frac{-1}{ab}\)

Step 1: Understand the Problem

We need to simplify the expression \(\frac{a^{-1} - b^{-1}}{a - b}\). This involves working with negative exponents and algebraic fractions.

Step 2: Rewrite Negative Exponents

Use the rule that \(a^{-1} = \frac{1}{a}\) and \(b^{-1} = \frac{1}{b}\). Rewrite the expression as \(\frac{\frac{1}{a} - \frac{1}{b}}{a - b}\).

Step 3: Find a Common Denominator

Combine the fractions in the numerator. The common denominator for \(\frac{1}{a}\) and \(\frac{1}{b}\) is \(ab\). So, rewrite the numerator as \(\frac{b - a}{ab}\).

Step 4: Simplify the Numerator

Substitute the new form of the numerator back into the original expression: \(\frac{\frac{b - a}{ab}}{a - b}\).

Step 5: Simplify the Fraction

Notice that \(b - a = - (a - b)\), so the expression becomes \(\frac{\frac{- (a - b)}{ab}}{a - b}\).

Step 6: Cancel Out Terms

Simplify by cancelling out \(a - b\) in the numerator and denominator: \(\frac{-1}{ab}\).

Step 7: Present the Final Answer

The simplified form of the expression is \(\frac{-1}{ab}\).

Key Concepts

Negative Exponents

Negative exponents can be confusing at first, but they are quite simple once you understand the basic rule. When you see an exponent as a negative, you can think of it as 'flipping' the base. For example, the expression \(a^{-1}\) is the same as \(\frac{1}{a}\). Similarly, \(b^{-1}\) means \(\frac{1}{b}\).
This helps in rewriting algebraic expressions and makes it easier to work with them. In our exercise, we changes \(a^{-1} - b^{-1}\) to \(\frac{1}{a} - \frac{1}{b}\).

Common Denominators

When you combine fractions, it's important to have a common denominator. This makes it possible to perform addition or subtraction between them. The common denominator is simply a value that both denominators can divide into.

• For the fractions \(\frac{1}{a}\) and \(\frac{1}{b}\), the common denominator is \(ab\), because both 'a' and 'b' can divide into \(ab\).
• Once you have a common denominator, you can rewrite the fractions accordingly. So, \(\frac{1}{a}\) becomes \(\frac{b}{ab}\) and \(\frac{1}{b}\) becomes \(\frac{a}{ab}\).

In the exercise, we use this principle to rewrite \(\frac{1}{a} - \frac{1}{b}\) as one single fraction: \(\frac{b - a}{ab}\).

Fraction Simplification

Simplifying fractions is all about making them look more manageable.

• In our exercise, we end up with the fraction \(\frac{\frac{b - a}{ab}}{a - b}\).
• Notice how the term \(b - a\) can be simplified by writing it as \(- (a - b)\).
• We replace \(b - a\) with \(- (a - b)\) and our fraction becomes \(\frac{\frac{- (a - b)}{ab}}{a - b}\).

This helps to cancel out the \(a - b\) term in the numerator and denominator.
The final step is making sure our fraction is as simple as possible. After cancellation, we are left with \(\frac{-1}{ab}\). This is the simplified form, where the fraction is in its most reduced state.