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Chapter 7: Problem 16

Simplify each complex fraction. Use either method. $$ \frac{\frac{3}{m}-m}{\frac{3-m^{2}}{4}} $$

Short Answer

Expert verified
\( \frac{4}{m} \)

Step by step solution

01

- Identify the Numerator and Denominator

The given complex fraction is \( \frac{\frac{3}{m}-m}{\frac{3-m^{2}}{4}} \). Here, the numerator is \( \frac{3}{m} - m \) and the denominator is \( \frac{3 - m^{2}}{4} \).
02

- Find a Common Denominator for the Numerator

Rewrite the numerator by finding a common denominator. The numerator is \( \frac{3}{m} - m \). The common denominator is \( m \). Rewrite \( m \) as \( \frac{m^2}{m} \). Thus, the numerator becomes \( \frac{3 - m^2}{m} \).
03

- Divide the Numerator by the Denominator

Since we have \( \frac{\frac{3 - m^2}{m}}{\frac{3 - m^2}{4}} \), this is the same as multiplying by the reciprocal of the denominator: \( \frac{3 - m^2}{m} \times\ \frac{4}{3 - m^2} \).
04

- Simplify the Expression

Simplify the expression \( \frac{3 - m^2}{m} \times\ \frac{4}{3 - m^2} \). The \( 3 - m^2 \) terms cancel out: \( \frac{4}{m} \). Thus, the simplified expression is \( \frac{4}{m} \).

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Key Concepts

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

Numerator and Denominator
Understanding the parts of a fraction is crucial. In any fraction, the top part is called the numerator and the bottom part is called the denominator.
For example, in the fraction \( \frac{a}{b} \), \( a \) is the numerator and \( b \) is the denominator.
In the complex fraction provided, \( \frac{\frac{3}{m}-m}{\frac{3-m^{2}}{4}} \), the numerator is everything above the division line: \( \frac{3}{m} - m \) and the denominator is everything below it: \( \frac{3 - m^2}{4} \). Identifying these parts correctly is essential before proceeding with any further steps.
Common Denominator
To simplify complex fractions, finding a common denominator for the fractions within the numerator or denominator helps to combine them easily.
For example, if we have \( \frac{3}{m} - m \), we need to rewrite these terms with a shared base.
Here's how:
  • Notice that in \( \frac{3}{m} \), the denominator is \( m \), but in \( m \), it's \( 1 \).
  • We convert \( m \) to \( \frac{m^2}{m} \) to have a common base \( m \).
  • So, \( \frac{3}{m} - m \) now becomes \( \frac{3}{m} - \frac{m^2}{m} = \frac{3 - m^2}{m} \).
Combining the terms this way is simple and makes the later steps much easier to follow.
Reciprocal Multiplication
Dividing fractions can be tricky, but there's a simple trick: multiply by the reciprocal!
For instance, if you have \( \frac{a}{b} \times \frac{c}{d} \), the reciprocal of \( \frac{c}{d} \) is \( \frac{d}{c} \). Multiplying by the reciprocal can turn a complex fraction into a simpler expression.
In our problem:
  • We have \( \frac{\frac{3 - m^2}{m}}{\frac{3 - m^2}{4}} \).
  • This can be rewritten by multiplying the numerator by the reciprocal of the denominator: \( \frac{3 - m^2}{m} \times \frac{4}{3 - m^2} \).
This step is essential and makes simplifying the expression straightforward by eliminating division and replacing it with multiplication by the reciprocal.
Canceling Terms
When simplifying fractions, canceling common terms helps to reduce the expression.
This works when both the numerator and the denominator share common factors.
In our example:
  • We have \( \frac{3 - m^2}{m} \times \frac{4}{3 - m^2} \).
  • Here, \( 3 - m^2 \) appears in both the numerator and the denominator.
  • We can cancel out \( 3 - m^2 \) because they are common terms.
This leaves us with \( \frac{4}{m} \).
By canceling out the common terms, we've transformed a complex fraction into a simple fraction that is easier to understand and work with.

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