Problem 22 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(k - 7\).

Step 1: Identify the common denominator

The common denominator in this expression is already given as both terms have the denominator of \(k+7\).

Step 2: Combine the numerators

Since the denominators are the same, the next step is to combine the numerators: \[ \frac{k^2}{k+7} - \frac{49}{k+7} = \frac{k^2 - 49}{k+7} \]

Step 3: Factor the numerator

The numerator \(k^2 - 49\) is a difference of squares, and can be factored as follows: \[ k^2 - 49 = (k + 7)(k - 7) \]

Step 4: Simplify the expression

Replace the factored form into the expression: \[ \frac{(k+7)(k-7)}{k+7} \] Since \(k+7\) is in both the numerator and the denominator, they cancel each other out, leaving: \[ k - 7 \]

Key Concepts

common denominator

When simplifying rational expressions, the first crucial step is to ensure the fractions have a common denominator.
This makes it easier to combine them into a single rational expression.
In the given exercise, both fractions already have a common denominator of \( k+7 \).
If the denominators were different, you would need to find an equivalent common denominator for both fractions.
This often involves finding the least common multiple (LCM) of the denominators.

combining numerators

Once you have a common denominator, you can combine the numerators into one fraction.
This is simply done by performing the operation indicated (addition or subtraction) on the numerators, keeping the common denominator in place.
In our exercise, the numerators are \( k^2 \) and \( 49 \) with a subtraction operation:

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

Combining the numerators is a key step in simplifying the expression.

difference of squares

The next step involves understanding and applying the concept of the difference of squares.
The difference of squares formula is:

• \( a^2 - b^2 = (a+b)(a-b) \)

In our exercise, the numerator \( k^2 - 49 \) is a difference of squares.
It can be written as:

• \( k^2 - 7^2 \)

Therefore, you can factor it into:

• \( (k+7)(k-7) \) This step simplifies the expression further.

factoring expressions

Factoring expressions is a crucial step while simplifying rational expressions.
You need to be vigilant in identifying patterns like the difference of squares, which simplifies complex expressions.
In our example, once we've identified the numerator \( k^2 - 49 \) as a difference of squares, we factor it into:

• \( (k+7)(k-7) \)

This converts the expression to:

• \( \frac{(k+7)(k-7)}{k+7} \)

Factoring simplifies the next steps considerably.

simplification

The final step is simplification, where we cancel common terms in the numerator and denominator.
In the simplified form of the expression:

• \( \frac{(k+7)(k-7)}{k+7} \)

The term \( k+7 \) appears in both the numerator and the denominator.
These terms cancel each other out, leaving:

• \( k-7 \)

The rational expression is now fully simplified.