Problem 44 - Prealgebra and Introductory Algebra

Question

Evaluate the variable expression for \(a=-2, b=4, c=-1,\) and \(d=3\) $$a^{2}-b^{2}$$

Step-by-Step Solution

Answer

-12

Step 1: Substitution of Variables

The first step is to replace the variables \(a\) and \(b\) in the expression with their given values. For this case, \(a=-2\) and \(b=4\). This substitution results in \((-2)^2 - (4)^2\).

Step 2: Evaluating the Expression

The next step is to simplify the expression and calculate the result. It is important to remember the precedence of operations where exponentiation comes first before subtraction. Therefore, \((-2)^2 = 4\) and \(4^2 = 16\). Hence, the expression becomes \(4 - 16\)

Step 3: Simplification

The final step involves the subtraction operation which yields the final solution of the problem. Thus, \(4 - 16 = -12\)

Key Concepts

Substitution of Variables

Substitution of variables is a foundational skill in algebra that involves replacing variables with their given numerical values. In the expression provided, which is \(a^2 - b^2\), we must substitute the variables \(a\) and \(b\) with the numbers assigned to them: specifically \(a = -2\) and \(b = 4\).

This process involves simply "plugging in" the numbers in place of their respective variables. Hence, the expression \(a^2 - b^2\) transforms into

• \((-2)^2 \) for \(a^2\)
• \((4)^2\) for \(b^2\)

After substitution, we arrive at the expression \((-2)^2 - (4)^2\). This step is crucial as it sets the stage for properly evaluating the expression using arithmetic operations.

Order of Operations

The order of operations is a set of rules that dictate the sequence in which operations should be performed to ensure consistent results. Often summarized by the acronym PEMDAS, it stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In our current problem, the focus lies on exponents and subtraction. According to the order of operations:

• First, we must handle any parentheses, though in this expression they only indicate the numbers getting squared.
• Then, we tackle exponentiation: compute \((-2)^2\) and \(4^2\) before proceeding with any subtraction.

By following these rules, we ensure the expression is evaluated correctly, which in this case involves calculating the squares before performing the subtraction.

Exponentiation

Exponentiation is a mathematical operation involving the raising of a number to a power. It tells us how many times to multiply a number by itself. In our exercise, we have

• \((-2)^2\)
• \((4)^2\)

This means:

\((-2)^2\) signifies multiplying -2 by itself, resulting in

• \((-2) imes (-2) = 4\)

Similarly, for \((4)^2\), it implies

• \(4 imes 4 = 16\)

Understanding how to work with exponents is essential for simplifying expressions, as it allows us to convert repeated multiplication into a straightforward operation.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This often means performing all necessary arithmetic operations to achieve a single numerical result. In our expression, after applying substitution and calculating the powers, we are left with:

• \(4 - 16\)

From here, we perform the subtraction: subtract 16 from 4.

The result of this operation is

• \(4 - 16 = -12\)

Thus, the simplified form of the original substituted expression is \(-12\). Simplification is the final step and ensures that the expression is expressed in the most concise and accurate form possible.