Problem 76 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(\frac{1}{2} n^{3}\) when \(n=-2\)

Step-by-Step Solution

Answer

The value of the expression \(\frac{1}{2} n^{3}\) when \(n=-2\) is -4

Step 1: Understand the expression

The given expression is \(\frac{1}{2} n^{3}\). This means you should cube the value of \(n\) and multiply the result by \(\frac{1}{2}\).

Step 2: Substitute \(n\) with \(-2\)

Replacing \(n\) with \(-2\) in the expression \(\frac{1}{2} n^{3}\), we will get \(\frac{1}{2} \times (-2)^{3} = \frac{1}{2} \times -8\).

Step 3: Perform the calculation

Performing the operation \(\frac{1}{2} \times -8\), we get -4. The result is -4 after performing the operation.

Key Concepts

Evaluating Expressions

Evaluating expressions simply means finding the numerical value of an algebraic expression. When we are given a mathematical expression, we often need to find out what the entire expression equals when the variables are replaced with actual numbers. For instance:

• You start by understanding what the expression describes.
• Identify the operations involved (such as addition, multiplication, or exponentiation).
• Follow the correct order of operations (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In this specific task, the expression is \[\frac{1}{2} n^{3}\],and we have to evaluate it when\[n = -2.\]By substituting and then solving, we can find the final value. Don't rush this process; double-check each step to ensure accuracy.

Exponents

Exponents are powerful mathematical tools that show how many times a number, called the base, is multiplied by itself. In the expression \(n^3\), the '3' is the exponent. It tells us to multiply \(n\) by itself three times:

• If \(n = -2\), then \((-2)^3 = (-2) \times (-2) \times (-2)\).
• First, calculate \((-2) \times (-2) = 4\). Notice that multiplying two negative numbers gives a positive result.
• Then, multiply again: \(4 \times (-2) = -8\). Here, multiplying a positive and a negative number results in a negative outcome.

So, for \((-2)^3\), the answer is \(-8\). Understanding how exponents work is key in algebra, especially when dealing with both positive and negative bases.

Substitution

Substitution in algebra involves replacing a variable with a given number to make it possible to calculate the expression's value. This process is essential because it helps transform an algebraic expression containing variables into a more straightforward numerical expression. Here's how it works:

• Take the original expression with its variable, \(\frac{1}{2} n^{3}\).
• Replace the variable \(n\) with the specified number, which in this case is \(-2\).
• Rewrite the expression with the number in place of \(n\): \(\frac{1}{2}\times (-2)^{3}\).

After substitution, you proceed with simplifying and solving the expression. This process of taking variables and changing them to numbers makes expressions concrete and actionable. Remember, substitution isn't just about changing symbols; it's about engaging deeply with the math to unravel the logic within.