Problem 24 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

\[ \frac{1}{k + 7} \]

Step 1: Identify the common denominator

Identify the common denominator in the expression. In this case, both fractions have the same denominator: \( k^{2} - 49 \). Notice that \( k^{2} - 49 \) can be factored as \( (k + 7)(k - 7) \).

Step 2: Combine the fractions

Since the fractions have the same denominator, they can be combined into a single fraction: \[ \frac{k - 7}{k^{2} - 49} \]

Step 3: Simplify the numerator

The combined numerator \( k - 7 \) can simply be written over the common denominator: \[ \frac{k - 7}{(k + 7)(k - 7)} \]

Step 4: Cancel out the common terms

Notice that \( k - 7 \) is a common factor in both the numerator and the denominator. Hence, you can cancel \( k - 7 \) from both: \[ \frac{1}{k + 7} \]

Key Concepts

Common Denominator

Before combining fractions, it is crucial to identify the common denominator. In this exercise, both fractions have the same denominator: \( k^{2} - 49 \). Having a common denominator means you can directly combine the fractions.
A good practice is to look for common denominators early. By doing this, you simplify the process later. It's worth noting that \( k^{2} - 49 \) can be factored as \((k + 7)(k - 7)\). This is a standard factoring technique you should remember. Make sure to always check if the denominators can be factored into simpler forms.

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into simpler multiplied factors. In this problem, we factor \( k^{2} - 49 \) as \((k + 7)(k - 7)\).
This step is essential because it simplifies the expression and makes it easier to cancel out common terms later. Always look out for common patterns like squares or differences of squares. For example:

• \(a^{2} - b^{2} = (a + b)(a - b)\) is a difference of squares

Practice factoring different types of polynomials to become quicker and more accurate.

Combining Fractions

To combine fractions, ensure they have the same denominator. In our example, the fractions already share the same denominator \(k^{2} - 49\).
When the denominators are the same, simply combine the numerators:

• \( \frac{k}{k^{2}-49} - \frac{7}{k^{2}-49} = \frac{k - 7}{k^{2}-49} \)

This process is straightforward because you only add or subtract the numerators and place them over the common denominator. Combining fractions simplifies the expression and sets you up for further simplification.

Canceling Common Terms

Canceling common terms is a powerful simplification technique. After combining the fractions, you often find common factors in the numerator and denominator.
In this example:

• \( \frac{k - 7}{(k + 7)(k - 7)} \)

Notice \(k - 7\) appears in both the numerator and denominator. These terms can be canceled out, leaving:

• \( \frac{1}{k + 7} \)

Always double-check to ensure you've canceled all possible terms. This final cancellation vastly simplifies the fraction, giving the most straightforward form of the expression.