Question

Simplify the expression. $$8\left(\frac{x}{8}\right)$$

Short answer

The simplified version of \(8\left(\frac{x}{8}\right)\) is \(x\).

Step-by-step solution

Distribute

The first step is to distribute the outside number to the inside of the parentheses. Here, that means multiplying 8 by each term inside the parentheses. Since the only term there is \(\frac{x}{8}\), this simplification process gives you \(8 \cdot \frac{x}{8}\).

Simplify

Multiply the numerator (the top part) of the fraction by 8. 8 times \(x\) is 8\(x\), which means the new numerator is 8\(x\). The denominator (the bottom part) of the fraction is still 8. This leaves you with \(\frac{8x}{8}\).

Divide

Now, divide the numerator by the denominator. \(8x\) divided by 8 simplifies to \(x\). The expression is thus simplified to \(x\).

Key concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term within a parenthesis. It’s like handing out a value evenly to each member inside the brace. For example, if we consider the expression \(a(b + c)\), we distribute \(a\) over \(b\) and \(c\), resulting in \(ab + ac\).

This property not only works with addition but also with subtraction: \(a(b - c) = ab - ac\). Understanding the distributive property is crucial as it lays the groundwork for expanding expressions, solving equations, and simplifying complex equations.

When we look at our exercise \(8(\frac{x}{8})\), we apply the distributive property. However, since there is only one term within the parentheses, it seems we’re distributing 8 over a single fraction. It might look redundant, but this step is essential in understanding the relationship between the number 8 outside and the denominator 8 within the fraction.

Numerators and Denominators

Understanding numerators and denominators is key to working with fractions. The numerator, located at the top of a fraction, represents how many parts we have. The denominator, at the bottom, tells us into how many equal parts the whole is divided. Consider the fraction \(\frac{3}{4}\): the numerator 3 indicates we have three parts, while the denominator 4 means the whole is divided into four equal parts.

In algebra, numerators and denominators can contain variables, such as \(\frac{x}{8}\) in our exercise. When we multiply this fraction by 8, we apply the operation only to the numerator, leading to \(\frac{8x}{8}\). Here, the relationship between the numerator and the denominator becomes visible, as we have identical numbers, indicating a potential simplification.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator and the denominator are as small as possible and have no common factors other than 1. A fraction is simplified completely when you cannot divide both the numerator and the denominator by the same number anymore.

A crucial aspect of simplifying is recognizing when a numerator and denominator can cancel each other out. If you have \(\frac{8x}{8}\), since 8 is a common factor in both the numerator and the denominator, they can be divided by 8, resulting in simplification to \(x\).

Simplifying can involve factoring numbers or variables to identify common factors. It’s a process that often makes addition, subtraction, multiplication, and especially division of fractions much more manageable. In our original exercise, once we distribute and multiply, we reach a fraction that displays clear indication for simplification – leading to the neat and clean result of just \(x\).