Problem 90 - Algebra 1

Question

Evaluate the expression. \(\frac{24}{x^{3}}\) when \(x=2\)

Step-by-Step Solution

Answer

The evaluated result of the expression \(\frac{24}{x^{3}}\) when \(x=2\) is 3.

Step 1: Understanding the expression

The given expression is \(\frac{24}{x^{3}}\) and it is required to evaluate this expression when \(x=2\). This involves substitution of the given 'x' value into the expression.

Step 2: Substitution

Now substitute \(x = 2\) into the expression. Upon substitution, the expression will become \(\frac{24}{2^{3}}\).

Step 3: Calculation

Next, calculate \(2^{3}\) which equals 8. The expression now is \(\frac{24}{8}\). Then calculate the division of 24 by 8 which gives 3.

Key Concepts

Evaluating Expressions

Evaluating expressions means using given values to calculate an expression's result. It involves understanding the components of the expression and performing mathematical operations. In our problem, the expression is \(\frac{24}{x^{3}}\). To evaluate it when \(x = 2\), we need to replace \(x\) with its given value and simplify. This process allows us to understand how changes in variables affect outcomes of expressions.

Substitution

Substitution is a core concept in algebra that involves replacing a variable with a specific value. It's like filling in the blanks in a sentence. In the given expression, \(\frac{24}{x^{3}}\), the variable \(x\) needs to be substituted with 2. By inserting 2 in place of \(x\), the expression becomes \(\frac{24}{2^{3}}\). This step sets the stage for further simplification. Substitution is critical for evaluating expressions accurately.

Exponents

Exponents are used to represent repeated multiplication of a number by itself. It simplifies expressions and calculations by showing a base number multiplied by itself a number of times. In the expression \(2^{3}\), the base is 2 and the exponent is 3, which means \(2 \times 2 \times 2\). Upon simplifying, \(2^{3}\) equals 8. Understanding exponents helps make complex expressions more manageable.

Division

Division is a basic arithmetic operation that involves splitting a number into equal parts. After substituting and calculating the exponent, the problem boils down to dividing 24 by 8, as seen in \(\frac{24}{8}\). Division provides the final result by determining how many times the divisor fits into the dividend. In this case, 24 divided by 8 equals 3. Mastering division is essential in solving many algebraic expressions efficiently.