Question

Multiply and reduce. Do some by calculator. $$\frac{x^{2}-a^{2}}{x y} \cdot \frac{x y}{x+a}$$

Short answer

\(x-a\)

Step-by-step solution

Identify a Difference of Squares

Recognize that the numerator of the first fraction, \(x^{2}-a^{2}\), is a difference of squares which can be factored into \((x+a)(x-a)\).

Factor and Simplify

Factor the difference of squares and rewrite the expression: \(\frac{(x+a)(x-a)}{xy} \cdot \frac{xy}{x+a}\).

Cancel Common Terms

Cancel out the common terms \((x+a)\) in the numerator of the first fraction and the denominator of the second fraction, and cancel out \(xy\) across the numerator and denominator.

Write the Result

After canceling, the only terms left are \((x-a)\) in the numerator and 1 in the denominator. So the reduced form of the expression is simply \(x-a\).

Key concepts

Difference of Squares

The concept of the difference of squares plays a pivotal role in simplifying algebraic expressions. It's based on the algebraic identity \( a^2 - b^2 = (a+b)(a-b) \), which shows that any expression where two squares are subtracted can be factored into the product of two distinct binomials. For instance, in our exercise \( x^2 - a^2 \) is identified as a difference of squares, which can be readily factored to \( (x+a)(x-a) \). Understanding this identity allows for swift simplification, especially when multiplying fractions involving algebraic terms.

Factorization

Factorization is the process of breaking down a complex expression into simpler, multipliable parts called factors. In the context of algebra, factorization often refers to breaking down polynomials into products of polynomials of lower degrees. The difference of squares is a classic case of factorization where we transform a quadratic polynomial into the product of two linear polynomials. In our example, \( x^2 - a^2 \) was factored into \( (x+a)(x-a) \). This not only helps to visualize the expression more clearly but sets up the stage for further simplification through cancellation of common terms.

Canceling Common Terms

When simplifying algebraic fractions, canceling common terms is a crucial step. This involves identifying and removing identical factors present in both the numerator and the denominator. Doing so simplifies the fraction to its lowest terms. In our exercise, after factorization, we observed that the term \( x+a \) was present in the numerator of the first fraction and the denominator of the second. Similarly, \( x y \) appeared in both the numerator and denominator when the two fractions were multiplied. These common terms were canceled out, significantly simplifying the expression to \( x-a \). Always remember to cancel only factors, not terms added or subtracted within an expression.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Just like numeric fractions, rational expressions can be simplified by canceling common factors in the numerator and the denominator. However, care must be taken to ensure these are actually factors and not terms separated by addition or subtraction. The simplified form of a rational expression is more straightforward and involves lesser degree polynomials, often making it easier to handle, especially when performing operations such as multiplication, division or solving equations. In our working example, we demonstrated how to simplify a rational expression by factoring and canceling, arriving at the simplest form, \( x-a \).