Question

Find the sum or difference. \(\frac{x}{16 x^2}-\frac{4}{16 x^2}\)

Short answer

\(\frac{x - 4}{16 x^2}\)

Step-by-step solution

Identify the Like Fractions

Both \(\frac{x}{16 x^2}\) and \(\frac{4}{16 x^2}\) are like fractions because they have the same denominator, i.e., \(16 x^2\).

Subtract the Numerators

Subtract the numerators of the like fractions while keeping the denominator the same. This gives: \(\frac{x - 4}{16 x^2}\).

Simplify the Fraction

The obtained fraction can't be simplified further. Therefore, the answer is \(\frac{x - 4}{16 x^2}\).

Key concepts

Understanding Like Fractions

When working with algebraic fractions, identifying 'like fractions' is a critical first step. These are fractions that have the same denominator, or bottom number. Imagine them as siblings from the same family—they share a common denominator and are therefore considered 'like' or similar. In simpler terms, if you have two or more fractions that have identical denominators, you are dealing with like fractions.

For example, in the exercise \(\frac{x}{16 x^2}-\frac{4}{16 x^2}\), both fractions have \(16 x^2\) as the denominator. This shared denominator is what makes them 'like fractions.' Working with like fractions is easier because you aren't required to do any fancy footwork with finding a common denominator before you can operate on them—you can simply proceed to the next step, which is manipulating the numerators.

Subtracting Numerators

Once you've established that you're working with like fractions, the next step is to 'subtract the numerators' if you're trying to find the difference between the fractions. Think of the numerators as pieces of a pie, and when you subtract, you're taking pieces away. It's important to keep the common denominator the same and only alter the numerators.

Let's dissect the example we have: \(\frac{x}{16 x^2} - \frac{4}{16 x^2}\). Both fractions share \(16 x^2\), so to find the difference, we simply subtract the second numerator from the first, resulting in a new fraction \(\frac{x - 4}{16 x^2}\). It's like saying, “I had \(x\) items, I gave away 4, now I have \(x - 4\) items left,” while you continue to carry the same box, \(16 x^2\), the entire time.

Algebraic Expressions Within Fractions

Algebraic expressions are the heart and soul of algebra—they combine numbers, variables (like x or y), and operation signs (like + or -) to represent a quantity. When these expressions become part of a fraction, they're treated much like numerical fractions, but with a twist: you must also consider the variables and apply algebra rules when simplifying.

For instance, in the algebraic fraction \(\frac{x - 4}{16 x^2}\), '\(x - 4\)' is an algebraic expression sitting at the top (numerator), and \(16 x^2\) is at the bottom (denominator). Simplifying algebraic fractions might involve factoring out common terms or reducing the expression to its simplest form. However, in this case, the fraction is already in its simplest form since there are no common factors to remove from the numerator and denominator. Hence, the importance of understanding algebraic expressions is paramount—they hold the key to unlocking the next level in algebraic manipulation and simplifying your work.