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Chapter 7: Problem 37

Add or subtract. Write answer in lowest terms. \(\frac{7}{3 p^{2}}-\frac{2}{p}\)

Short Answer

Expert verified
\frac{7 - 6p}{3p^2}

Step by step solution

01

Find a common denominator

The common denominator for the fractions \(\frac{7}{3p^2}\) and \(\frac{2}{p}\) is \3p^2\.
02

Rewrite each fraction with the common denominator

Rewrite \(\frac{2}{p}\) so it has the common denominator \3p^2\. Multiply the numerator and the denominator of \(\frac{2}{p}\) by 3p to get \(\frac{2 \times 3p}{p \times 3p} = \frac{6p}{3p^2}\).
03

Perform the subtraction

Now perform the subtraction: \(\frac{7}{3p^2} - \frac{6p}{3p^2} = \frac{7 - 6p}{3p^2}\).
04

Simplify the expression if possible

The expression \(\frac{7 - 6p}{3p^2}\) is already in its simplest form since the numerator and the denominator have no common factors.

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Key Concepts

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

Common Denominator
When adding or subtracting fractions, finding a common denominator is crucial. A common denominator is a shared multiple of the denominators of the given fractions.
This allows us to rewrite the fractions so that they have the same base, enabling easy addition or subtraction.
For example, in the exercise \(\frac{7}{3p^2} - \frac{2}{p}\), the common denominator is \(3p^2\).
This is because \(3p^2\) is the least common multiple of the denominators \(3p^2\) and \(p\).
Identifying the common denominator simplifies the process of fraction operations.
Fraction Subtraction
Once we have a common denominator, we can proceed with fraction subtraction. This involves rewriting each fraction so they share the same denominator.
In the given example, \(\frac{2}{p}\) is converted to \(\frac{6p}{3p^2}\).
This is achieved by multiplying the numerator and the denominator by the missing factor, which is \(3p\) in this case.
The next step is to subtract the numerators while keeping the common denominator:
\(\frac{7}{3p^2} - \frac{6p}{3p^2} = \frac{7 - 6p}{3p^2}\).
Ensure the subtraction is correctly performed and that both fractions are accurately represented with the common denominator before combining.
Simplifying Fractions
Simplifying fractions involves reducing them to their lowest terms. This means making sure that the numerator and the denominator have no common factors other than 1.
After performing the subtraction, we end up with \(\frac{7 - 6p}{3p^2}\).
To check if this fraction can be simplified further, we examine the numerator \(7 - 6p\) and the denominator \(\frac{3p^2}\).
In this case, they have no common factors, meaning the fraction is already in its simplest form.
Always verify if a fraction can be simplified by looking for common factors or using prime factorization if necessary.
Simplifying fractions makes the final answer cleaner and easier to interpret.

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Most popular questions from this chapter

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