Problem 42 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(3(-4)(n)\) when \(n=-2\)

Step-by-Step Solution

Answer

The value of the expression \(3(-4)(n)\) when \(n=-2\) is 24.

Step 1: Substitute the Value of \(n\)

Replace \(n\) in the expression \(3(-4)(n)\) with -2. So, the expression becomes \(3(-4)(-2)\).

Step 2: Apply the Rule of Signs

Apply the rule of signs. The product of two negative numbers is positive. Therefore, the new expression is \(3 \times 4 \times 2\).

Step 3: Perform the Multiplication

Now, perform the multiplication: \(3 \times 4 \times 2 = 24\).

Key Concepts

Substituting Values into Expressions

Substituting values into expressions is a fundamental skill in algebra. It involves replacing variables with given numbers. In the original exercise, we are asked to evaluate the expression \( 3(-4)(n) \) when \( n = -2 \). This process is known as substitution. It transforms the expression with a variable into one with numbers only. Here's how it works simply:

• Identify the variable in the expression, which is \( n \) in this case.
• Replace \( n \) with the value it's given, which is \(-2\).

After substitution, our expression \( 3(-4)(n) \) becomes \( 3(-4)(-2) \). This sets the stage for the next steps in solving the equation.In any algebraic problem involving expressions with variables, ensure that you substitute correctly. Double-checking this step is crucial since it changes the expression completely.

Rule of Signs in Multiplication

A key principle to remember in multiplication, especially when involving negative numbers, is the rule of signs. This rule states:

• The product of two negative numbers is positive.
• The product of a positive number and a negative number is negative.

In our exercise, once we substituted \( n \), we got the expression \( 3(-4)(-2) \). According to the rule of signs, we need to multiply the two negative numbers first:

• \((-4) \times (-2) = 8\)

The product is a positive \( 8 \) because the rule states that multiplying two negatives results in a positive outcome. This changes our expression to \( 3 \times 8 \). Remembering the rule of signs helps ensure the accuracy of your calculations, especially when dealing with multiple negative numbers.

Multiplying Integers

Once you've substituted values and applied the rule of signs, you're ready to tackle multiplication. Multiplying integers is straightforward when you know the basic multiplication facts. For our problem, we ended with the expression \( 3 \times 8 \).Here's how to proceed:

• Simply multiply the numbers: \( 3 \times 8 = 24 \).

This gives us our final answer of \( 24 \).When doing multiplication:

• Start from left to right or multiply numbers in parts if it simplifies the problem.
• Keep track of any sign changes based on the rule of signs.

Practicing with integers ensures you get faster and more accurate. Whether it's with small numbers or large, the method remains consistent. It's a skill that fundamentally supports your ability to work through algebraic expressions.