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Chapter 7: Problem 21

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

Short Answer

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

Step by step solution

01

Identify the common denominator

Both terms in the expression have the same denominator, which is \(w + 8\).
02

Combine the numerators

Since the denominators are already the same, we can combine the numerators: \(w^2 - 64 \). This gives us: \[\frac{w^{2} - 64}{w + 8}\].
03

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: \[\frac{(w - 8)(w + 8)}{w + 8}\].
04

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: \[w - 8\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=4\).

For exercises 61-64, the completed problem has one mistake. (a) Describe the mistake in words or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: In the formula \(A=\frac{10}{B}\), is the relationship between \(A\) and \(B\) a direct variation or an inverse variation? Incorrect Answer: Since as \(B\) increases, \(A\) also increases, this is a direct variation.

The relationship of the distance driven, \(x\), and the cost of gasoline, \(y\), is a direct variation. For a trip of \(250 \mathrm{mi}\), the cost is \(\$ 90\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost of gasoline to drive \(225 \mathrm{mi}\). d. What does \(k\) represent in this equation?

For exercises \(67-82\), use the five steps and a proportion. In \(2010,3.5\) per 100,000 full-time equivalent workers were killed on the job with a total of 547 workers killed on the job. Find the number of full-time equivalent workers used to create this ratio. Round to the nearest whole number. (Source: www.osha.gov)

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