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Chapter 7: Problem 5

In the following exercises, simplify each rational expression. $$ -\frac{44}{55} $$

Short Answer

Expert verified
-\frac{4}{5}

Step by step solution

01

Identify the Numerator and Denominator

The given rational expression is \(-\frac{44}{55}\). Identify the numerator (\(-44\)) and the denominator (\(55\)).
02

Find the Greatest Common Divisor (GCD)

Determine the greatest common divisor (GCD) of the numerator and the denominator. For \(44\) and \(55\), the GCD is \(11\).
03

Divide both the Numerator and Denominator by the GCD

Divide the numerator and the denominator by their GCD:\(-\frac{44}{55} = -\frac{44 \div 11}{55 \div 11} = -\frac{4}{5}\).

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Key Concepts

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

greatest common divisor
Understanding the greatest common divisor (GCD) is essential in fraction simplification. The GCD of two numbers is the largest number that divides both of them without a remainder. To find the GCD, follow these steps:
  • List the factors of each number.
  • Identify the common factors.
  • Select the largest common factor.
In our example, for the numbers 44 and 55, the common factors are 1 and 11. Therefore, the GCD is 11. This helps to simplify the fraction by ensuring both the numerator and the denominator can be divided evenly by the highest possible number.
numerator and denominator
In any fraction, the numerator is the top number which represents how many parts you have. The denominator is the bottom number which shows how many equal parts the whole is divided into. For instance, in \[-\frac{44}{55}\], -44 is the numerator and 55 is the denominator.

Understanding the roles of the numerator and denominator is crucial for fraction simplification. You always divide the numerator by the same number you divide the denominator to maintain the value of the fraction.
fraction simplification
Simplifying fractions means reducing them to their simplest form. This involves dividing both the numerator and the denominator by their GCD. The goal is to get the smallest possible whole numbers that represent the same value.

Let鈥檚 recap the steps with our example, \[-\frac{44}{55}\]:
  • Identify the GCD of 44 and 55, which is 11.
  • Divide both the numerator (-44) and the denominator (55) by 11.
  • The simplified fraction is \[-\frac{4}{5}\].
It is important to simplify fractions to make mathematical operations easier and to provide a clearer representation of the value.
intermediate algebra
Intermediate algebra often involves working with rational expressions. Simplifying these expressions requires understanding concepts like the GCD, numerators and denominators, and fraction simplification.

In more advanced problems, you might deal with variables and polynomials. However, the process of simplification remains the same. Identify the GCD, factor out the terms, and reduce the expression to its simplest form. Mastering these steps with basic numerical fractions prepares you for more complex algebraic tasks.

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