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Chapter 3: Problem 33

Simplify each expression. $$\frac{5 \pi}{4}-\frac{\pi}{2}$$

Short Answer

Expert verified
\( \frac{3 \pi}{4} \)

Step by step solution

01

Identify Common Denominator

To simplify the expression \( \frac{5 \pi}{4} - \frac{\pi}{2} \), first identify a common denominator for the fractions. The denominators are 4 and 2. The least common denominator is 4.
02

Rewrite Fractions With Common Denominator

Rewrite \( \frac{\pi}{2} \) so it has the same denominator as \( \frac{5 \pi}{4} \). Multiply both the numerator and the denominator of \( \frac{\pi}{2} \) by 2 to get \( \frac{2 \pi}{4} \). Now the expression is: \( \frac{5 \pi}{4} - \frac{2 \pi}{4} \).
03

Subtract the Numerators

Now that the fractions have the same denominator, subtract the numerators: \( 5 \pi - 2 \pi = 3 \pi \). Therefore, the expression becomes: \( \frac{3 \pi}{4} \).

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Key Concepts

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

Least Common Denominator
When dealing with algebraic fractions, identifying the least common denominator (LCD) is crucial for simplifying expressions involving subtraction or addition. The least common denominator is the smallest number that is a multiple of each of the denominators in the problem. In our example, we have two denominators, 4 and 2. The LCD will be the smallest number that both these numbers can divide into evenly.
For 4 and 2:
  • 2 divides into 4, making 4 a multiple of both denominators
  • Hence, the least common denominator is 4
Identifying this helps us to rewrite each fraction with a common denominator, which is the first necessary step in the subtraction process.
Subtracting Fractions
To effectively subtract fractions, their denominators must be the same. Once the least common denominator is identified and fractions are rewritten with this common denominator, subtracting fractions becomes straightforward.
Here's how it works:
  • Write each fraction with the least common denominator. For \( \frac{\pi}{2} \), multiply both the numerator and denominator by 2, thus obtaining \( \frac{2\pi}{4} \).
  • Now the fractions are \( \frac{5\pi}{4} \) and \( \frac{2\pi}{4} \).
  • With the same denominator, you subtract the numerators directly: \( 5\pi - 2\pi = 3\pi \).
The operation focuses solely on the numerators since the denominators are already aligned.
Simplifying Expressions
After you've subtracted the numerators, the resulting fraction may still need to be simplified. Simplifying expressions involves reducing the fraction to its simplest form. However, in some cases, the expression is already in its simplest form.
For the fraction \( \frac{3\pi}{4} \):
  • The numerator is \(3\pi\), and the denominator does not contain \(\pi\)
  • This expression is already simplified as neither the numerator nor the denominator have common factors besides 1.
  • Thus, \( \frac{3\pi}{4} \) is the simplified result of the subtraction \( \frac{5\pi}{4} - \frac{\pi}{2} \).
Always check to ensure there's no further simplification possible, ensuring your answer is as simple as it can be.

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