Problem 7 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{x}{x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of \( \frac{x}{x^2} \) is \( \frac{1}{x} \).

Step 1: Identify Numerator and Denominator

The numerator of the given fraction is \( x \), and the denominator is \( x^2 \). We need to simplify this fraction.

Step 2: Factor Common Terms

The fraction \( \frac{x}{x^2} \) can be written as \( \frac{x}{x \cdot x} \), factoring the denominator to show common terms.

Step 3: Cancel Common Factors

Since both the numerator and denominator have the common factor \( x \), we can cancel \( x \) out from both. This gives us \( \frac{1}{x} \).

Step 4: Check Simplification

The fraction \( \frac{1}{x} \) no longer has any common factors between the numerator and denominator, so it is in its simplest form.

Key Concepts

Numerator and Denominator

Fractions are a way to represent parts of a whole. Each fraction is made up of two main parts: the numerator on top and the denominator on the bottom. The numerator represents how many parts you have, while the denominator tells you the total number of equal parts in the whole. For example, in the fraction \( \frac{3}{4} \), the numerator is 3, and the denominator is 4.
In the given exercise, the fraction is \( \frac{x}{x^2} \). Here, the numerator is \( x \), and the denominator is \( x^2 \), which can also be seen as \( x \times x \). Understanding these components is crucial, as they determine how we proceed with simplifying the fraction.

Cancel Common Factors

When simplifying a fraction, one of our main goals is to reduce it to its simplest form. This involves canceling common factors present in both the numerator and the denominator. A factor is a number or variable that evenly divides another number or expression.
Let's see how this works using the fraction \( \frac{x}{x^2} \). If we break down the denominator as \( x \cdot x \), we notice both the numerator and one of the parts of the denominator have \( x \) as a common factor. We can then 'cancel' this \( x \), which is like dividing both the numerator and the denominator by \( x \).
After canceling, we are left with \( \frac{1}{x} \). This step is essential in simplifying fractions, ensuring they are displayed in their simplest and most understandable form. Simplifying makes it easier to perform further calculations and understand the fraction at a glance.

Simplest Form

The simplest form of a fraction is when we have reduced it in such a way that there are no common factors between the numerator and the denominator other than 1. This means the fraction is as compact as possible without altering its value.
In the fraction \( \frac{x}{x^2} \), once we have canceled the common \( x \), we are left with \( \frac{1}{x} \). This fraction now has no further common factors between the numerator and the denominator since the numerator is 1. Remember, 1 is a universal factor and doesn't count in simplifying terms further.
Writing a fraction in its simplest form makes working with it much more straightforward, whether it is for adding, subtracting, multiplying, or dividing fractions. Always aim for the simplest form when dealing with fractions for optimal understanding and ease in computation.