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Chapter 8: Problem 290

In the following exercises, simplify. $$ \frac{7+\frac{2}{q-2}}{\frac{1}{q+2}} $$

Short Answer

Expert verified
The simplified form is \[ 7q + 14 + \frac{2q + 4}{q-2} \].

Step by step solution

01

- Simplify the Complex Fraction

First, recognize the complex fraction. The main fraction is \( \frac{7+\frac{2}{q-2}}{\frac{1}{q+2}} \). Start by simplifying the numerator and the denominator separately.
02

- Rewrite the Denominator

Notice that the denominator \( \frac{1}{q+2} \) can be rewritten as \( \frac{1}{q+2} = (q+2)^{-1} \).
03

- Simplify the Entire Fraction

Simplify the entire fraction by multiplying the numerator by the reciprocal of the denominator: \[ \frac{7 + \frac{2}{q-2}}{\frac{1}{q+2}} = (7 + \frac{2}{q-2}) \times (q+2) \].
04

- Distribute and Combine Terms

Distribute \( q+2 \) across the terms in the numerator: \[ (7 + \frac{2}{q-2})(q+2) = 7(q+2) + \frac{2}{q-2}(q+2) \].
05

- Final Simplification

Combine terms: \[ 7q + 14 + \frac{2(q+2)}{q-2} = 7q + 14 + \frac{2q + 4}{q-2} \]. The terms cannot be simplified further without knowing more about \( q \).

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Key Concepts

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

Simplifying Complex Fractions
Complex fractions can be intimidating, but they become simpler with the right approach. Essentially, a complex fraction is a fraction where the numerator, denominator, or both are also fractions. Let's break it down:

Take the example: \[\frac{7+\frac{2}{q-2}}{\frac{1}{q+2}}\] The main fraction has smaller fractions within it.
Step-by-step simplification:
  • Recognize the parts. Here, the numerator is \(7 + \frac{2}{q-2}\), and the denominator is \(\frac{1}{q+2}\).
  • Simplify each part independently. Notice that the denominator \(\frac{1}{q+2}\) can be transformed using the reciprocal.
  • Combine them. We multiply by the reciprocal to eliminate the fraction in the denominator.
This method helps to simplify and make sense of seemingly complex expressions
Understanding the Reciprocal
Reciprocals are key tools when working with complex fractions. The reciprocal of a number or fraction essentially flips it. For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).

For our problem, the denominator is \( \frac{1}{q+2} \). The reciprocal of this is \( q+2 \).
  • Reciprocals simplify division. Multiplying by a fraction's reciprocal is the same as dividing by the fraction.
  • In our example, multiplying the numerator \(7+\frac{2}{q-2}\) by \(q+2\) simplifies the fraction.
It's important to understand this concept, as it turns a complex fraction into a simpler multiplication problem.
Applying the Distributive Property
The distributive property is a foundational mathematical principle stating that for any numbers \a\, \b\, and \c\, \(a(b + c) = ab + ac\). It allows the distribution of multiplication over addition or subtraction within parentheses.

Applying this to our fraction:
  • \( (7 + \frac{2}{q-2})(q+2) \) expands to \( 7(q+2) + \frac{2}{q-2}(q+2) \).
  • This means distributing \(q+2\) to both terms inside the parentheses.
  • Applying this property makes combining terms straightforward.
This property is valuable in simplifying expressions and combining like terms effectively. For our exercise, it bridges the gap between recognizing the fraction and arriving at its simplest form.

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