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Chapter 7: Problem 25

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

Short Answer

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

Step by step solution

01

Identify the common denominator

Notice that both terms in the expression have the same denominator, which is \(w^2 + 64\).
02

Combine the fractions

Since the denominators are identical, you can combine the numerators over the common denominator. This gives: \[ \frac{w - 8}{w^2 + 64} \]
03

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.

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Key Concepts

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

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.

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Most popular questions from this chapter

For exercises 1-10, (a) solve. (b) check. $$ \frac{4}{15} k+\frac{3}{4}=-2 $$

For exercises 49-52, the formula \(C=\frac{P_{m} P_{i}}{T F}\) describes the cost of insurance, \(C\). Is the relationship of the given variables a direct variation or an inverse variation? $$ C, T, \text { and } F \text { are constant; the relationship of } P_{i} \text { and } P_{m} $$

For exercises \(41-44\), the formula \(R=\frac{V C}{T}\) describes the flow rate of fluid \(R\) through an intravenous drip. Is the relationship of the given variables a direct variation or an inverse variation? $$ V \text { and } C \text { are constant; the relationship of } R \text { and } T \text {. } $$

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)

For exercises \(67-82\), use the five steps and a proportion. In 2010 , there were \(14.9\) cases of syphilis per 100,000 Americans with a total of 45,834 cases of syphilis. Find the population of Americans used to create this ratio. Round to the nearest hundred. (Source: www.cdc.gov, 2011)
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