Problem 21 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{w^{2}}{w+8}-\frac{64}{w+8} $$

Step-by-Step Solution

Answer

The simplified form is \(w - 8\).

Step 1: Identify the common denominator

Both terms in the expression have the same denominator, which is \(w + 8\).

Step 2: Combine the numerators

Since the denominators are already the same, we can combine the numerators: \(w^2 - 64 \). This gives us: \[\frac{w^{2} - 64}{w + 8}\].

Step 3: Factor the numerator

The numerator \(w^2 - 64\) is a difference of squares, which can be factored into \((w - 8)(w + 8)\). Therefore, we get: \[\frac{(w - 8)(w + 8)}{w + 8}\].

Step 4: Cancel out the common factor

The factor \(w + 8\) in the numerator and denominator can be canceled out, leaving us with: \w - 8\. Thus, the simplified form is: \[w - 8\].

Key Concepts

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. In our exercise, both terms have the same denominator, which is \(w + 8\). This makes it easier to simplify the expression. If the denominators were different, we would have to find a common denominator before combining the terms.

Difference of Squares

The difference of squares is a special algebraic pattern where two perfect squares are subtracted. This pattern is represented as \(a^2 - b^2 = (a - b)(a + b)\). In our problem, we identified the numerator \(w^2 - 64\) as a difference of squares. Here, \(w^2\) is \(a^2\) and \(64\) is \(b^2\). Therefore, we can factor \(w^2 - 64\) as \( (w - 8)(w + 8)\). This step is crucial for further simplification.

Factoring

Factoring is the process of breaking down an expression into simpler 'factors' that, when multiplied together, give the original expression. In the exercise, we factored the numerator \(w^2 - 64\) into \( (w - 8)(w + 8)\). This step helps to make the expression easier to work with and sets up the next steps in the simplification process.

Canceling Common Factors

Canceling common factors is a simplification step where we eliminate identical factors in the numerator and denominator. In our exercise, the factor \(w + 8\) appears in both the numerator and denominator. By canceling this common factor, we simplify the expression \(\frac{(w - 8)(w + 8)}{w + 8}\) to \(w - 8\). This leaves us with a much simpler expression, making it easier to understand and use in further calculations.