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Chapter 4: Problem 9

Reduce each of the following rational expressions to lowest terms. $$\frac{-15}{-6}$$

Short Answer

Expert verified
The reduced form of \(\frac{-15}{-6}\) is \(\frac{5}{2}\).

Step by step solution

01

- Identify the Rational Expression

The given rational expression is \(\frac{-15}{-6}\).
02

- Simplify Signs

Both numerator and denominator are negative, so the signs will cancel each other. Thus, \(\frac{-15}{-6}\) simplifies to \(\frac{15}{6}\).
03

- Find the Greatest Common Divisor (GCD)

Find the greatest common divisor of 15 and 6. The GCD of 15 and 6 is 3.
04

- Divide Numerator and Denominator by the GCD

Divide both the numerator and the denominator by the GCD (which is 3): \(\frac{15 \div 3}{6 \div 3} = \frac{5}{2}\).
05

- Express in Lowest Terms

The rational expression \(\frac{15}{6}\) in its lowest terms is \(\frac{5}{2}\).

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Key Concepts

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

greatest common divisor
The greatest common divisor (GCD) is a crucial concept when simplifying rational expressions. It helps us find the largest number that divides both the numerator and the denominator without leaving a remainder. To find the GCD of two numbers:
  • List the factors of each number. For example, factors of 15 are 1, 3, 5, and 15. Factors of 6 are 1, 2, 3, and 6.
  • Identify the common factors. Here, 1 and 3 are common.
  • Choose the largest factor that is common. In this case, it is 3.
This GCD helps reduce the fraction to its simplest form by dividing both the numerator and denominator by it.
fraction simplification
Simplifying a fraction, or a rational expression, means reducing it to its lowest terms. This occurs when the numerator and the denominator have no common factors other than 1. Here's a step-by-step easy method:
  • Find the GCD of the numerator and denominator.
  • Divide both the numerator and the denominator by this GCD.
For instance, simplifying \(\frac{15}{6}\) involves finding the GCD (which is 3) and then dividing: \(\frac{15 梅 3}{6 梅 3} = \frac{5}{2}\). This provides the simplest form of the rational expression.
negative signs in fractions
Handling negative signs in fractions can be tricky, but it's quite straightforward if you follow the rule that a negative sign can be moved around the fraction. This means \( \frac{-a}{b} \), \( \frac{a}{-b} \), and \( -\frac{a}{b} \) are all equivalent. Here are steps to simplify:
  • If both numerator and denominator are negative, they cancel each other out, leaving a positive result.
  • If only one is negative, the fraction remains negative, and you can place the negative sign either in the numerator, the denominator, or in front of the fraction.
In our example, \( \frac{-15}{-6} \) simplifies to \( \frac{15}{6} \), and then further reduction yields \( \frac{5}{2} \).

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Most popular questions from this chapter

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{5}{18 a^{3}}-\frac{1}{90 a^{2}}$$

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. Takima has a semester average of \(94 .\) She has four exam scores of \(88,85,93\) and \(95 .\) If her final exam counted for 3 exams, find her score on the final.

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{11 c}{60 a^{2} b}-\frac{2 c}{45 a b^{3}}$$

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. Sal has \(\$ 3200\) invested in an account paying \(7.2 \%\) interest per year. At what rate must he invest an additional \(\$ 2800\) so that the annual interest from the two investments will be \(\$ 390 ?\)

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. If the temperature range on a certain day was between \(25^{\circ} \mathrm{C}\) and \(40^{\circ} \mathrm{C},\) what was the temperature range in degrees Fahrenheit? Use the formula \(C=\frac{5}{9}(F-32)\)
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