Problem 77 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ x 2-y 2(x-y) 2 $$

Step-by-Step Solution

Answer

x + y

Step 1: Expand the expression

We start with the expression x 2 - y 2(x - y) 2. This expression is likely a result of confusion in formatting. Assume you meant (x^2 - y^2) / (x-y)^2. To continue, first recognize that x^2 - y^2 can be factored as a difference of squares: (x + y)(x - y).

Step 2: Simplify using cancelling common terms

Once you factor x^2 - y^2 to (x + y)(x - y), notice that the (x-y) term in the numerator can be cancelled with the (x-y)^2 term in the denominator. This reduces the expression to (x + y) / (x-y).

Step 3: Simplify the expression

After canceling the common (x-y) terms from the numerator and denominator, we end up with x + y. Thus, the simplified expression is x + y.

Key Concepts

Difference of Squares

In algebra, the difference of squares is a special kind of polynomial factorization. It is one of the first factorization techniques students learn because it helps to simplify many expressions. The difference of squares technique applies to expressions of the form \( a^2 - b^2 \). This particular form represents two perfect squares separated by a minus sign.

The reason why it’s called 'difference of squares' is because \( a^2 \) and \( b^2 \) are both squares of some numbers or variables. Understanding this concept allows us to factor such expressions using the formula:

• \( a^2 - b^2 = (a + b)(a - b) \)

This formula shows that the original expression can be rewritten as the product of two binomials: one with a sum and the other with a difference of the same terms. For example, in our original exercise, the term \( x^2 - y^2 \) is a difference of squares which can be factored into \( (x + y)(x - y) \). Recognizing and applying this pattern is a crucial step in simplifying algebraic expressions.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing a complex expression into a simpler or more manageable form. The goal of simplification is to make the expression easier to understand and work with.

When you simplify expressions, you often look to:

• Factor out common terms or elements
• Combine like terms
• Reduce fractions or expressions by canceling common factors

In the original exercise, after recognizing the difference of squares, the expression \((x^2 - y^2) / (x-y)^2\) is simplified by factoring the numerator. By writing \(x^2 - y^2\) as \((x + y)(x - y)\), we prepare the expression for further simplification. This step is crucial before you can move on to cancel similar terms, making the expression easier to interpret at a glance.

Canceling Terms

Canceling terms is a simplifying technique used when both the numerator and denominator of a fraction or fraction-like expression have a common factor. This process can transform a complex expression into a simpler one, making calculations or further evaluations more straightforward.

To cancel terms:

• Look for identical factors in both the numerator and denominator
• Divide both numerator and denominator by this common factor
• Simplify the expression that remains

In our example, after factoring \(x^2 - y^2\) as \((x + y)(x - y)\), the common term \((x - y)\) appears both in the numerator and as \((x - y)^2\) in the denominator. By canceling \((x - y)\) from both, we reduce the expression to \((x + y)/(x - y)\) and further to just \(x + y\). This revolutionary step reduces complexity and leads directly to the simplified solution, validating the process of factoring and canceling common terms in algebra.