Problem 78 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(\frac{24}{x^{3}}\) when \(x=2\)

Step-by-Step Solution

Answer

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

Step 1: Substitute the given value into the expression

Replace \(x\) in the expression \(\frac{24}{x^{3}}\) with the given value 2. This gives us \(\frac{24}{2^{3}}\).

Step 2: Follow the order of operations

From the order of operations, the exponent of 3 on number 2 (i.e., \(2^3\)) should be calculated first. So, it becomes \(\frac{24}{8}\).

Step 3: Perform the division

After computing the exponent, the expression simplifies to \(\frac{24}{8}\) which performs to 3.

Key Concepts

Order of Operations

When working with algebraic expressions, applying the correct order of operations is crucial to ensure the right solution is reached.
The order of operations is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.

This means:

• First, tackle any calculations inside parentheses
• Next, evaluate the exponents
• Then, perform multiplication and division from left to right as they appear in the expression
• Finally, handle addition and subtraction from left to right

In this exercise, after substituting the value for the variable, you will first compute the exponentiation before moving on to division.
Remember, multiplication and division are of equal precedence, as are addition and subtraction, hence why they're both tackled in sequence from left to right.

Substitution

Substitution is a simple but powerful technique where we replace a variable in an expression with a given number.
This method allows us to evaluate expressions for specific values.

In this exercise, the variable is substituted by the number 2.
Here's how you do it:

• Identify the variable in the expression
• Replace the variable every time it appears with the given value, which is 2 in this instance

This leads to the expression \( \frac{24}{2^3} \).Substitution helps transform the algebraic expression into a numerical one, which you then solve using arithmetic operations.

Exponentiation

Exponentiation is a mathematical operation where a number, called the base, is multiplied by itself a specified number of times, which is denoted by the exponent.
For example, in the term \(2^3\), 2 is the base and 3 is the exponent, which calculates to 2 × 2 × 2.

This results in 8.
In our problem, following the substitution step where we got \( \frac{24}{2^3} \), the exponentiation needs to be calculated first before moving to the division.

• The exponentiation step gives \(2^3 = 8\).
• Subsequently, you divide 24 by this result to obtain the final value

Thus, understanding how to work with exponents allows you to simplify expressions accurately and efficiently, ultimately reaching the right conclusion.