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Chapter 4: Problem 2

Reduce each of the following rational expressions to lowest terms. $$\frac{-8}{12}$$

Short Answer

Expert verified
\frac{-8}{12} reduces to \frac{-2}{3}.

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To reduce the rational expression \(\frac{-8}{12}\) to its lowest terms, first identify the greatest common divisor (GCD) of the numerator (-8) and the denominator (12). The GCD of -8 and 12 is 4.
02

Divide the Numerator by the GCD

Divide the numerator -8 by the GCD (4). This results in \(\frac{-8}{4} = -2\).
03

Divide the Denominator by the GCD

Divide the denominator 12 by the GCD (4). This results in \(\frac{12}{4} = 3\).
04

Write the Reduced Expression

Combine the results from the previous two steps to write the reduced rational expression. The reduced form of \(\frac{-8}{12}\) is \(\frac{-2}{3}\).

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Key Concepts

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

greatest common divisor
When you want to reduce a rational expression, the greatest common divisor (GCD) is your best friend.
The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder.
Imagine you have the fraction \(\frac{-8}{12}\). To reduce this fraction, we first need to determine the GCD of -8 and 12.
Both -8 and 12 are divisible by 4. By dividing -8 and 12 by 4, the GCD, you can simplify your fraction significantly.
Understanding the GCD can make reducing fractions much simpler and clearer.
numerator and denominator
Let's talk about the numerator and the denominator, the two key parts of any fraction.
In our example, the numerator is -8, and the denominator is 12.
The numerator is the top part of the fraction, representing how many parts of the whole we have.
The denominator is the bottom part, indicating into how many parts the whole is divided.
For the rational expression \(\frac{-8}{12}\), we are essentially looking at -8 parts out of a total of 12. This is why understanding both parts is crucial before we simplify the fraction.
simplifying fractions
Simplifying fractions is all about making the fraction as simple as possible while still representing the same value.
The process involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example, with \(\frac{-8}{12}\), we found out that the GCD is 4.
Dividing the numerator -8 by the GCD, we get -2. Dividing the denominator 12 by the GCD, we get 3.
So, \(\frac{-8}{12}\) simplifies to \(\frac{-2}{3}\). Simplifying helps in making the calculations easier and the fraction more understandable.

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Most popular questions from this chapter

Solve the equations and inequalities. $$0.06 x+0.04(5.000-x)=320$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. The length of a rectangle is 1 less than twice the width. If the width is 1 more than one seventh of the perimeter, find the dimensions of the rectangle.

solve the equations and inequalities. \(0.05 x+0.08(x+3,000)=890\)

Discuss what is wrong with each of the following: (a) \(\frac{3 x^{2}}{2 x}=\frac{3}{2 x}\) (b) \(\frac{5 x}{25 x^{2}} =\frac{5 x}{5 \cdot 5 x x}=\frac{0}{5 x}=0\) (c) \(\frac{2 x y}{6 x^{3} y^{2}} \stackrel{2}{2 \cdot 3 x y x^{2} y} \stackrel{2}{=3 x^{2} y}\) (d) \(\frac{3 x+2 x^{2}}{x} \stackrel{\text {=}}{} \frac{3 x+2 x^{2}}{x}=3+2 x^{2}\)

Solve the equations and inequalities. $$\frac{x+3}{5}-x=2$$
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