Problem 16 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{8}{12 x-16 y}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{2}{3x - 4y} \).

Step 1: Identify the Numerator and the Denominator

The algebraic fraction \( \frac{8}{12x - 16y} \) has 8 as the numerator and \( 12x - 16y \) as the denominator. Our goal is to simplify this fraction by reducing it to its simplest form.

Step 2: Find the Greatest Common Factor (GCF)

Identify the greatest common factor for both the numerator and the terms in the denominator. The common factor for 8, 12, and 16 is 4.

Step 3: Factor Out the GCF

Divide both the numerator and each term of the denominator by the GCF, 4: \( \frac{8}{12x - 16y} = \frac{8 \div 4}{(12x \div 4) - (16y \div 4)} = \frac{2}{3x - 4y} \).

Step 4: Verify the Simplified Expression

Ensure that \( \frac{2}{3x - 4y} \) is the simplest form. Check for any remaining common factors, but since none exist, \( \frac{2}{3x - 4y} \) is the simplest form.

Key Concepts

Greatest Common Factor

In mathematics, the greatest common factor (GCF) plays a crucial role in simplifying algebraic expressions. The GCF is the largest number that evenly divides two or more numbers. Knowing how to find the GCF helps make algebraic fractions easier to work with by reducing them to their simplest form.

When you look at the fraction \( \frac{8}{12x - 16y} \), finding the GCF of the numbers in the expression allows us to simplify it efficiently. In this case:

• The numerator is 8.
• The terms in the denominator are 12 and 16.

The greatest common factor of 8, 12, and 16 is 4. By dividing both the numerator and each term of the denominator by this factor, you simplify the fraction considerably. This step makes it easier to work with and is critical in algebra.

Numerator and Denominator

Understanding the roles of the numerator and the denominator is key to simplifying fractions. In any fraction, the numerator is the number or expression above the fraction line, and the denominator is below. They are often expressions that tell us about ratios or division of specific quantities.

For the algebraic fraction \( \frac{8}{12x - 16y} \):

• 8 is the numerator. It's a constant and the number being divided.
• \( 12x - 16y \) is the denominator. It is a polynomial expression that indicates the division of 8 across the reduced expression of 12 and 16, which are multiplied by variables \( x \) and \( y \).

Recognizing each part of the fraction helps in determining how to proceed with simplification. The goal is to ensure that the fraction is reduced in a way that the numerator and denominator no longer have any common factors except for 1.

Simplest Form

The simplest form of an algebraic fraction means reducing it so no further simplification can be done. This involves ensuring that the numerator and the denominator share no common factors other than 1. Simplifying to the simplest form makes fractions much easier to interpret and work with in broader algebra problems.

In the process of simplifying the fraction \( \frac{8}{12x - 16y} \):

• We first factor out the greatest common factor (GCF), which is 4.
• Dividing both the numerator and each term of the denominator by 4 results in \( \frac{2}{3x - 4y} \).

Once verified, \( \frac{2}{3x - 4y} \) is in its simplest form, indicating that no further reduction is possible. Thus, understanding and applying the concept of simplest form is essential in correctly simplifying algebraic fractions.