Question

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

Short answer

\(\frac{(b+a)(b-a)}{a^{2}b^{2}}\)

Step-by-step solution

Find a common denominator

The common denominator of the two fractions \(\frac{1}{a^{2}}\) and \(\frac{1}{b^{2}}\) is \(a^{2}b^{2}\). To combine the fractions, each fraction must be rewritten with this common denominator.

Rewrite the fractions with the common denominator

Rewrite each fraction with the common denominator, which gives us \(\frac{b^{2}}{a^{2}b^{2}}\) for the first fraction and \(\frac{a^{2}}{a^{2}b^{2}}\) for the second fraction. This results in \(\frac{b^{2}}{a^{2}b^{2}} - \frac{a^{2}}{a^{2}b^{2}}\).

Combine the fractions

Now that the fractions have the same denominator, simply subtract the numerators to combine them into a single fraction: \(\frac{b^{2} - a^{2}}{a^{2}b^{2}}\).

Factor the difference of squares in the numerator

The numerator is a difference of squares, which can be factored using the identity \(x^{2} - y^{2} = (x + y)(x - y)\). Apply this identity to the numerator to get \((b + a)(b - a)\). Thus, the expression becomes \(\frac{(b+a)(b-a)}{a^{2}b^{2}}\).

Key concepts

Common Denominator

Finding a common denominator is a fundamental step in the process of adding, subtracting, or comparing fractions. A common denominator refers to a shared multiple of the denominators of two or more fractions. To make mathematical operations possible, each fraction is converted to an equivalent fraction with this shared denominator, making calculations straightforward.

Consider our example with \(\frac{1}{a^{2}}\) and \(\frac{1}{b^{2}}\). Since the denominators are \(a^2\) and \(b^2\), their least common multiple is \(a^{2}b^{2}\), giving us a platform to combine the fractions harmoniously. It's like finding a common language for two people who speak different languages, enabling them to communicate effectively.

Difference of Squares

The difference of squares is a special pattern in algebra that emerges when we subtract one square number from another. This pattern can be identified by the formula \(x^{2} - y^{2} = (x + y)(x - y)\).

In the context of our exercise, once we combine the fractions with the common denominator, we are left with the numerator \(b^{2} - a^{2}\), which is a difference of squares. By recognizing this pattern, we are able to factor the numerator into \(b + a)\) and \(b - a\), simplifying the expression and revealing its underlying structure. It's akin to splitting a beautifully symmetrical sculpture into two equally complementary parts, each reflecting the other.

Factoring Algebraic Expressions

Factoring is essentially the reverse of multiplication in algebra. It involves breaking down a complex expression into simpler components or factors that, when multiplied together, give back the original expression.

In our exercise, after finding a common denominator and combining the fractions, we encounter the numerator \(b^{2} - a^{2}\). Factoring this difference of squares is a powerful tool that transforms this complex expression into a product of binomials, \(b + a)\) and \(b - a\). Factoring can simplify expressions, solve equations, and provide insights into the properties of algebraic expressions, much like a skilled artist deconstructing a complex scene into basic shapes and forms.

Rational Expressions Simplification

Simplifying rational expressions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. The aim is to make the expression as clear and as concise as possible.

In the solution to our exercise, after combining like fractions and factoring the difference of squares, we do not simplify the expression further as there are no common factors to divide out in the numerator \((b + a)(b - a)\) and the denominator \(a^{2}b^{2}\). However, if there were common factors, we would proceed to cancel them out, stripping the expression to its bare essentials. Simplifying rational expressions makes them easier to work with; it's analogous to cleaning up and organizing a workspace for maximum efficiency and clarity.