Problem 25 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \( \frac{w - 8}{w^2 + 64} \).

Step 1: Identify the common denominator

Notice that both terms in the expression have the same denominator, which is \(w^2 + 64\).

Step 2: Combine the fractions

Since the denominators are identical, you can combine the numerators over the common denominator. This gives: \[ \frac{w - 8}{w^2 + 64} \]

Step 3: Simplify the expression

Check if the numerator \(w - 8\) and the denominator \(w^2 + 64\) can be factored further or simplified. In this case, they cannot be simplified any further.

Key Concepts

Common Denominator

When dealing with rational expressions, a common denominator is necessary for combining fractions. Think of the denominator as the bottom part of the fraction. It tells you how many equal parts the whole is divided into. When two or more fractions have the same denominator, they are ready to be combined. In our exercise, the denominator for both fractions is \(w^2 + 64\). This makes it straightforward to combine the fractions, as they already share a common base. Remember, without a common denominator, you must first find one before combining fractions.

Combining Fractions

Once you've identified a common denominator, combining fractions becomes simple. In our case, the expression is \( \frac{w}{w^{2}+64}-\frac{8}{w^{2}+64} \). Because both fractions share the denominator \(w^2 + 64\), we keep that denominator and combine the numerators. So, we merge the numerators \( w \) and \( -8 \). This process gives us one fraction:

• \(\frac{w - 8}{w^2 + 64}\)

This new fraction combines everything into a single fraction over the shared denominator, simplifying future steps.

Numerator and Denominator Simplification

The last step involves simplifying the numerator and the denominator. In our example, we have the combined fraction

• \(\frac{w - 8}{w^2 + 64}\)

To simplify, we check if there are any common factors or if the numerator and denominator can be factored further. Here, the numerator \(w - 8\) and the denominator \(w^2 + 64\) cannot be simplified any further because they do not share common factors. Evaluating whether expressions can be more simplified is a crucial step. If they can't, you've completed the process, and the expression is fully simplified.