Problem 59 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$ (8 x)^{3} \text { when } x=2 $$

Step-by-Step Solution

Answer

The value of the expression \(\frac{24}{x}\) when \(x = 3\) is 8.

Step 1: Understand the Expression

The given expression is \(\frac{24}{x}\). Here \(x\) is a variable and we are required to evaluate this expression when \(x = 3\). This variable substitution is a standard method in algebra problem solving.

Step 2: Substitute the Given Value

Substitute the given value \(x = 3\) into the expression: \(\frac{24}{x}\) turns into \(\frac{24}{3}\).

Step 3: Simplifying the Expression

Complete the division operation. \(\frac{24}{3} = 8\)

Key Concepts

Variable Substitution

When working with algebraic expressions, one of the first skills to master is variable substitution. This process involves replacing a variable with a specific value that is provided or that we choose. In our example, we are asked to evaluate the expression \(\frac{24}{x}\) for a given value of \(x\), which is 3.

It's vital to understand that the variable, in this case, \(x\), is simply a placeholder for a number. When we substitute \(x\) with 3, the abstract expression \(\frac{24}{x}\) becomes a concrete numerical expression \(\frac{24}{3}\) that we can compute. This is how we transform unknowns into knowns, allowing us to solve for the required quantity.

Simplifying Expressions

Once we have substituted the variable with its value, the next step is simplifying the expression. Simplification typically involves performing any arithmetic operations such as addition, subtraction, multiplication, or division. In our case, after substituting 3 for \(x\), we get \(\frac{24}{3}\), which is a division operation.

To simplify this expression, we divide the numerator (the top number) by the denominator (the bottom number). The goal of simplifying is to rewrite the expression in an easier-to-understand form; for \(\frac{24}{3}\), simplification would yield the result of 8. Remember, a simplified expression should be as straightforward as possible, so there’s no room for confusion.

Division Operation

In algebra and basic arithmetic, the division operation is one of the four fundamental operations. It's the process of determining how many times one number is contained within another. When we see the division symbol, which can be a forward slash (/), as in our example, or a horizontal bar in fraction notation, we understand it to mean 'divided by'.

For our expression \(\frac{24}{3}\), we are essentially asking, 'how many times does 3 go into 24?' After performing the division, we find that the answer is 8, since 3 multiplied by 8 equals 24. Understanding the division operation is crucial, as it will be frequently encountered in various forms throughout algebra and advanced mathematics.