Problem 41 - Algebra 1

Question

SIMPLIFYING RATIONAL EXPRESSIONS Simplify the expression. $$\frac{8}{4 a+1}-\frac{5}{4 a+1}$$

Step-by-Step Solution

Answer

The simplified expression is: \(\frac{3}{4a+1}\)

Step 1: Identify the problem

Here, the problem is to simplify a rational expression. The given expression is \(\frac{8}{4a+1} - \frac{5}{4a+1}\). Both terms in the expression share the same denominator (\(4a+1\)).

Step 2: Use the properties of fractions

According to fraction properties, when two fractions have the same denominator, we can combine them into one fraction by performing the operation on the numerators. Therefore we subtract the numerator of the second fraction from the numerator of the first fraction, giving us: \(\frac{8 - 5}{4a+1}\).

Step 3: Subtract the numerators

Subtracting 5 from 8 gives us 3. Hence, our result is \(\frac{3}{4a+1}\).

Key Concepts

Rational Expression

A rational expression is much like a fraction, but instead of integers or real numbers in its numerator and denominator, it may contain variables. Just as fractions represent division of numbers, rational expressions represent division of polynomials. Simplifying them requires an understanding of fraction rules and can involve factoring, reducing and cancelling common factors, and the set of all numbers for which the expression is defined (its domain).

Consider the expression \(\frac{8}{4a+1} - \frac{5}{4a+1}\). This is a textbook example of a rational expression, comprising numerators 8 and 5, and a shared denominator \(4a+1\). When simplifying, always look for common denominators as they simplify the process—similar to how we combine simple fractions with a common denominator without altering the denominator itself.

Properties of Fractions

The properties of fractions are the backbone of working with rational expressions. Since rational expressions can be viewed as 'fractions with variables', the same rules apply. One key property to remember is that if two fractions share the same denominator, we can perform addition or subtraction on their numerators without changing the denominator. In other words, \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\) or \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\).

This property is central when dealing with equations like the one in our exercise. Since the given rational expressions share \(4a+1\) as their denominator, we are able to subtract their numerators directly, leading to a single, simplified rational expression \(\frac{3}{4a+1}\). Understanding these properties will enable students to effortlessly navigate more complex rational expressions.

Combine Like Terms

When simplifying expressions, it is crucial to combine like terms. Like terms are terms that contain the same variables raised to the same power. Combining them involves performing the required operations (addition, subtraction, multiplication, etc.) on their coefficients. For example, in the expression \(2x + 5x\), \(2x\) and \(5x\) are like terms, and their combination yields \(7x\).

In our rational expression scenario, while \(8\) and \(5\) may not seem like traditional 'terms with variables', they are, in a sense, like terms because they share the same denominator. Hence, we carry out the subtraction \(8 - 5\), and this arithmetic operation simplifies the rational expression's 'numerator' while keeping its 'denominator' intact. Recognizing this can greatly simplify the process of reducing rational expressions and is a skill that will prove valuable in managing more elaborate algebraic tasks.