Question

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

Short answer

The simplified product of the two fractions is \( \frac{ab}{x^2 - y^2} \).

Step-by-step solution

Multiply the numerators together

The first step in multiplying two fractions is to multiply the numerators together. The numerator of the product is the product of the numerators from the two given fractions. In this case, the numerators are 'a' and 'b'. Multiply 'a' by 'b' to get 'ab'.

Multiply the denominators together

Just as with the numerators, multiply the denominators together. In this case, multiply '(x - y)' by '(x + y)' to get '(x - y)(x + y)'.

Apply the difference of squares

The expression '(x - y)(x + y)' is a difference of squares, which can be simplified to 'x^2 - y^2'.

Combine the expression

Combine the simplified numerator and denominator to form the simplified expression of the multiplication. The final simplified expression is 'ab' over 'x^2 - y^2'.

Key concepts

Difference of Squares

The difference of squares is a common algebraic pattern which emerges when you are multiplying two binomials that are identical except for the sign between their terms. It is expressed as \( a^2 - b^2 \) when you have \( (a + b)(a - b) \).

Understanding the difference of squares is essential when you encounter problems that require you to multiply binomials with opposing signs, as seen in the exercise where you multiply \( (x - y)(x + y) \). When simplified, it results in \( x^2 - y^2 \), because the middle terms cancel out. This will always be true for any two terms \( a \) and \( b \) where \( a = x \) and \( b = y \).

Remember, the difference of squares is useful in simplifying expressions and can make complex multiplication problems much more manageable.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression more manageable by combining like terms, applying algebraic identities like the difference of squares, or reducing fractions. The goal is to transform a complex expression into a simpler form that's easier to understand or use in further calculations.

The process begins by looking for any recognizable algebraic patterns, such as the difference of squares we discussed previously. By identifying and applying these patterns, you can often reduce the number of terms or the complexity of the terms in the expression.

For example, in the exercise's product \( ab \) over \( x^2 - y^2 \) we've applied the difference of squares to simplify the denominator. No further reduction is possible as there are no common factors between the numerator and denominator. Understanding how to identify opportunities to simplify expressions is a fundamental skill in algebra that aids in solving equations and interpreting solutions.

Fraction Multiplication Steps

Multiplying fractions involves several clear steps, which, if followed correctly, will lead to the correct product. Here's a breakdown of the steps mentioned in the solution:

Step 1: Multiply the Numerators

Start by multiplying the top numbers, or numerators, of the fractions. For instance, multiplying \( a \) and \( b \) gives you \( ab \).

Step 2: Multiply the Denominators

Next, multiply the bottom numbers, or denominators. With expressions like \( (x - y) \) and \( (x + y) \) in the exercise, this yields \( (x - y)(x + y) \).

Step 3: Simplify

The last step is to simplify, if possible. Apply algebraic shortcuts like the difference of squares for the binomials in the denominator, reducing them to \( x^2 - y^2 \).

Always remember to check if the final fraction can be reduced further by finding common factors. In the given exercise, we end with \( ab \) over \( x^2 - y^2 \), which is already in its simplest form. These steps, when done sequentially, ensure clarity in the multiplication process and help avoid mistakes.