Problem 17 - Precalculus

Question

Evaluate each expression, given that \(a=-2\) \(b=3,\) and \(c=-4\). $$|a-b|^{2}$$

Step-by-Step Solution

Answer

The expression evaluates to 25.

Step 1: Substitute the Values

Replace the variables with the given numbers in the expression. Substitute \(a = -2\) and \(b = 3\) into the expression \(|a-b|^2\). This gives: \(|-2 - 3|^2\).

Step 2: Calculate the Expression Inside the Absolute Value

Perform the subtraction inside the absolute value. Calculate \(-2 - 3 = -5\). This gives us \(|-5|^2\).

Step 3: Evaluate the Absolute Value

Find the absolute value of \(-5\). The absolute value of \(-5\) is \(5\). So, we have \(5^2\).

Step 4: Square the Result

Finally, compute the square of \(5\). Calculate \(5^2 = 25\). Therefore, the expression evaluates to \(25\).

Key Concepts

Substitution

Substitution is a fundamental technique in algebra and mathematics as a whole. This involves replacing variables with their given numerical values within an expression or equation. It serves as an initial step to simplify and eventually solve the expression. Consider the expression \(|a-b|^{2}\). When we substitute the given values \(a = -2\) and \(b = 3\), the expression becomes \(|-2 - 3|^{2}\).
This step is crucial because it translates abstract representations (i.e., variables) into concrete numbers that we can operate on. Without substitution, solving or evaluating expressions with variables would be impossible when specific numerical outcomes are desired.

• Make sure to carefully replace each variable with the correct value to avoid errors.
• Maintain the integrity of the mathematical operations surrounding the variables to ensure accurate results.

Squaring

Squaring is a mathematical operation where a number is multiplied by itself. This operation is denoted by the superscript \(2\), as seen in expressions like \(x^{2}\). In our example, once the absolute value operation is completed, the squaring operation needs to be carried out on the result.
After substituting and simplifying, we found that \(|-5|\) is equal to \(5\). The next step involves squaring this absolute value, which means computing \(5 imes 5\). This results in \(25\).
Squaring is important in many mathematical contexts and often appears in formulas involving areas, the Pythagorean theorem, and quadratic equations. Here are a few quick reminders about squaring:

• Squaring is always a non-negative result, as multiplying two identical numbers (either both positive or both negative) leads to a positive outcome.
• Squaring effectively emphasizes the growth rate of numbers, as small increments lead to larger resultant values (e.g., \(7^2 = 49\) is notably larger than 7).

Expression Evaluation

Expression evaluation is the comprehensive process of calculating an expression's value through systematic operations such as substitution, squaring, and more. Each step in this process builds upon the last to reach a final, simplified conclusion.
In our particular problem, we begin by substituting the given values into the expression, which results in \(|-2 - 3|^{2}\). Then, we perform the arithmetic inside the absolute value: \(-2 - 3 = -5\).
The next move involves taking the absolute value to transform \(-5\) into its positive counterpart, \(5\). Finally, we square this result, \(5^2 = 25\), to get the evaluated expression.
With expression evaluation, it is critical to maintain the correct order of operations, also known as PEMDAS (parentheses, exponents, multiplication and division, addition and subtraction). This ensures that each phase of the evaluation reflects accurate and logical mathematical reasoning. Ultimately, expression evaluation provides us with a singular, calculable result from a potentially complex network of operations.