Problem 63 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{5}{12 x}-\frac{11}{16 x^{2}}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{20x - 33}{48x^2}\).

Step 1: Identify the Problem

We are given two fractions: \(\frac{5}{12x}\) and \(\frac{11}{16x^2}\). We need to subtract the second fraction from the first and then simplify the result.

Step 2: Find a Common Denominator

The denominators in the fractions are \(12x\) and \(16x^2\). The least common denominator (LCD) must be the smallest expression that both denominators will divide into, which is \(48x^2\).

Step 3: Rewrite Fractions with Common Denominator

Convert each fraction so that it has \(48x^2\) as the denominator. - For \(\frac{5}{12x}\), multiply both the numerator and the denominator by \(4x\) to get \(\frac{20x}{48x^2}\).- For \(\frac{11}{16x^2}\), multiply both the numerator and the denominator by \(3\) to get \(\frac{33}{48x^2}\).

Step 4: Subtract the Fractions

With a common denominator, subtract the numerators: \(\frac{20x}{48x^2} - \frac{33}{48x^2} = \frac{20x - 33}{48x^2}\).

Step 5: Simplify the Result

The result \(\frac{20x - 33}{48x^2}\) is already in its simplest form because there are no common factors in the numerator and the denominator that can be reduced further.

Key Concepts

Understanding Fractions

Fractions are a fundamental concept in mathematics, and they represent a part of a whole. Each fraction consists of two main parts:

• The numerator: the number on top, representing how many parts we have.
• The denominator: the number on the bottom, indicating how many equal parts the whole is divided into.

For instance, in the fraction \(\frac{3}{4}\), the numerator is 3, and the denominator is 4, which means we have 3 out of 4 equal parts.
When working with fractions, especially in algebra, it's crucial to understand how to manipulate and operate with them, such as finding common denominators or simplifying them. This understanding lays the groundwork for solving more complex algebraic expressions.

Finding the Least Common Denominator

The Least Common Denominator (LCD) is essential when adding or subtracting fractions. It is the smallest expression or number that each of the denominators can evenly divide into.
To find the LCD for algebraic fractions, follow these steps:

• Identify all the factors in each denominator.
• Determine the largest set of factors needed to cover all denominators.

In our example, the denominators are \(12x\) and \(16x^2\). We decompose these:

• \(12x = 2^2 \times 3 \times x\)
• \(16x^2 = 2^4 \times x^2\)

The LCD must include the highest power of each factor present. Therefore, the LCD is \(2^4 \times 3 \times x^2 = 48x^2\). Picking the LCD ensures that all fractions involved can be expressed with the same denominator, simplifying addition or subtraction.

Simplification of Algebraic Expressions

Simplification of algebraic expressions involves reducing expressions to their simplest form. This is crucial for making calculations easier and understanding the relationships within the expression.
The process typically involves:

• Combining like terms, which are terms that have the same variable raised to the same power.
• Reducing fractions by dividing the numerator and denominator by their greatest common divisor (GCD).
• Writing expressions with positive exponents and a standard form.

In our problem, after subtracting the rewritten fractions \[\frac{20x}{48x^2} - \frac{33}{48x^2} = \frac{20x - 33}{48x^2}\]we check if the numerator and the denominator have any common factors to simplify further. In this case, there aren't any common factors, so the fraction \(\frac{20x - 33}{48x^2}\) is in its simplest form. Simplification helps us find the most straightforward representation of an algebraic expression, ensuring it is as easy to interpret and use as possible.