Problem 26 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{k}{k^{2}+49}-\frac{7}{k^{2}+49} $$

Step-by-Step Solution

Answer

\[ \frac{k - 7}{k^{2}+49} \]

Step 1: Identify Common Denominator

Notice that both fractions have the same denominator: \(k^2 + 49\). So, the fractions \(\frac{k}{k^{2}+49}\) and \(\frac{7}{k^{2}+49}\) can be combined easily.

Step 2: Combine the Numerators

Since we have a common denominator, we can combine the numerators: \[ \frac{k}{k^{2}+49} - \frac{7}{k^{2}+49} = \frac{k - 7}{k^{2}+49} \]

Step 3: Simplify the Result

The simplified form of the given expression is: \[ \frac{k - 7}{k^{2}+49} \]

Key Concepts

Common Denominator

In algebra, a common denominator is essential when adding or subtracting rational expressions. A common denominator is a shared multiple of the denominators of two or more fractions. Without a common denominator, algebraic fractions cannot be combined.
Step 1 in the given exercise demonstrates the importance of the common denominator by identifying that both fractions have the same denominator: \(k^2 + 49\). This common denominator allows the fractions to be combined seamlessly.
When working with rational expressions, always look for a common denominator first. If the denominators are not the same, you might need to find the least common multiple (LCM) of the denominators to proceed.
Here, you don't need to find the LCM since \(k^2 + 49\) is already shared by both fractions. This makes combining the fractions straightforward.

Combining Numerators

Once you've identified the common denominator, the next step is to combine the numerators. In algebra, if the denominators are the same, you can subtract or add the numerators directly while keeping the common denominator unchanged.
The exercise already has a common denominator, so we can combine the numerators of the two fractions: \[ \frac{k}{k^{2}+49} - \frac{7}{k^{2}+49} = \frac{k - 7}{k^{2}+49} \] Combining the numerators means performing the operation indicated (addition or subtraction) on the numerators and placing the result over the common denominator.

• Always ensure the denominator stays the same
• Perform the arithmetic carefully on the numerators

In this case, the operation is subtraction, leading to \(k - 7\) as the new numerator.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying rational expressions makes them easier to work with. The exercise provided is a good example of simplifying a rational expression.
A rational expression might initially look complicated, but with steps like finding a common denominator and combining numerators, you can simplify it efficiently.
Here, we started with two fractions: \( \frac{k}{k^{2}+49} \) and \( \frac{7}{k^{2}+49} \), simplifying them to \( \frac{k-7}{k^{2}+49} \).
Simplifying rational expressions requires:

• Identifying common denominators
• Combining numerators or denominators appropriately
• Simplifying further by factoring if possible

By following these steps, it becomes much easier to work with and understand rational expressions in algebra.