Problem 42 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{a^{-1}-b^{-1}}{a^{-2}-b^{-2}}$$

Step-by-Step Solution

Answer

The simplified version of the given expression is \(-ab\).

Step 1: Rewrite with Positive Exponents

Replace \(a^{-1}\) with \(1/a\) and \(a^{-2}\) with \(1/a^2\) and do the same for the terms involving \(b\). This gives: \[\frac{1/a-1/b}{1/a^2-1/b^2}\]

Step 2: Simplify Fractions in the Numerator and Denominator

To subtract fractions, they must have a common denominator. In this case, the common denominator for the numerator is \(ab\) and that for the denominator is \(a^2b^2\). Use these to find equivalent fractions which can be subtracted. This gives:\[\frac{\frac{b-a}{ab}}{\frac{b^2-a^2}{a^2b^2}}\]

Step 3: Simplify the Entire Fraction

A fraction divided by another fraction is the same as multiplying the first fraction by the reciprocal of the second. Thus, rewrite as follows: \[\frac{b-a}{ab} \cdot \frac{a^2b^2}{b^2-a^2}\]This simplifies to:\[-ab\] Please note the change in sign because whenever a difference of two squares, in this case \(b^2-a^2\), is simplified, it results in \(-(a-b)\), not \(b-a\).

Key Concepts

Exponents and Powers

When dealing with algebra, exponents and powers can often appear, and understanding these helps to simplify expressions efficiently. An exponent tells us how many times we multiply a number by itself. For instance, in the expression \(a^n\), \(n\) is the exponent and \(a\) is the base. If \(n = 2\), then \(a^2 = a \cdot a\).

Working with powers means understanding how to manipulate these exponents. For example, when multiplying like bases, we add the exponents: \(a^m \cdot a^n = a^{m+n}\). On the other hand, when dividing, we subtract them: \(\frac{a^m}{a^n} = a^{m-n}\).

This becomes especially useful when you're rewriting expressions to simplify, as seen in the original problem. It's crucial to handle powers correctly to get anticipated simplification and accurate results when altering expressions that include them.

Fractions in Algebra

Fractions often complicate algebraic expressions, but with practice, they can be managed easily. A fraction consists of a numerator and a denominator. When subtracting fractions, it's essential that they have a common denominator. For instance,

• In the expression \(\frac{a^{-1} - b^{-1}}{a^{-2} - b^{-2}}\), convert to \(\frac{1/a - 1/b}{1/a^2 - 1/b^2}\).
• The common denominator for \(1/a - 1/b\) is \(ab\), enabling rewriting to \((b-a)/ab\).
• In the denominator, use \(a^2b^2\) as a common base, resulting in \((b^2-a^2)/a^2b^2\).

This step-by-step method underlines the importance of creating equivalent fractions to simplify algebraic expressions, ensuring simplification is both easy and logical.

Negative Exponents

Negative exponents often confuse students, but they are a simple reciprocal of positive exponents. A negative exponent indicates that the base is on the wrong side of the fraction bar and needs to be flipped. For example,

• \(a^{-1}\) becomes \(1/a\) when rewritten.
• Similarly, \(a^{-2}\) is rewritten as \(1/a^2\).

Understanding this conversion helps transition expressions to their simplest form. For example, converting negative exponents to positive ones in the initial problem makes subsequent steps clear and straightforward. Perhaps the most crucial point about negative exponents is never fearing them! Just remember they reflect division in disguise. This way, when you encounter \(a^{-n}\), you can confidently change it to \(1/a^n\), making your work with algebra much simpler.