Problem 18 - Precalculus

Question

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

Step-by-Step Solution

Answer

The expression evaluates to -8.

Step 1: Substitute Values

Start by substituting the given values of \(a\), \(b\), and \(c\) into the expression \(a^2 - |bc|\). Here, \(a = -2\), \(b = 3\), and \(c = -4\). The expression becomes \((-2)^2 - |3 \times (-4)|\).

Step 2: Calculate the Square

Calculate \((-2)^2\). The square of \(-2\) is \((-2) \times (-2) = 4\). Thus, \((-2)^2 = 4\).

Step 3: Calculate the Product

Next, calculate the product \(b \times c\), which is \(3 \times (-4) = -12\).

Step 4: Evaluate Absolute Value

Find the absolute value of the product calculated in Step 3. The absolute value of \(-12\) is 12, so \(|3 \times (-4)| = 12\).

Step 5: Complete the Expression

Substitute the results from Step 2 and Step 4 into the expression: \(4 - 12\). Simplify this to find the final value: \(4 - 12 = -8\).

Key Concepts

Substitution in Expressions

Substitution in expressions is a fundamental technique in algebra and arithmetic that simplifies solving problems by replacing variables with their known values. In our original exercise, we are given specific values for the variables \(a\), \(b\), and \(c\). We need to substitute these into the given expression \(a^2 - |bc|\). This step transforms the abstract expression into a concrete calculation.

• Why substitute? Substitution removes uncertainty by turning an equation with variables into one with numbers, making it easier to calculate.
• How to substitute: Each variable is replaced with its given numeric value. For instance, replace \(a\) with \(-2\), \(b\) with \(3\), and \(c\) with \(-4\).
• Example: The original expression \(a^2 - |bc|\) becomes \((-2)^2 - |3 \times (-4)|\) after substitution.

Understanding substitution is crucial for solving more complex expressions and equations.

Absolute Value

Absolute value is a concept in mathematics that represents the non-negative value of a number, regardless of its sign. In simpler terms, it's how far a number is from zero, on the number line, without considering the direction. This concept is frequently used when solving arithmetic expressions to simplify calculations involving negative numbers.Consider the expression part \(|3 \times (-4)|\):

• The product of \(3\) and \(-4\) equals \(-12\).
• The absolute value of \(-12\) is \(12\), represented as \(|-12| = 12\).

The absolute value ensures we use a positive number in our calculations, which assists in avoiding mistakes with subtraction and negative numbers. Therefore, understanding and correctly applying absolute value is essential in manipulating and simplifying expressions.

Basic Algebra

Basic algebra is the branch of mathematics that deals with simplifying expressions and solving equations involving variables. It forms the foundation for understanding more advanced mathematics.In the context of our exercise, basic algebra involves:

• Applying arithmetic operations: This includes addition, subtraction, multiplication, and raising numbers to a power, as seen when we calculate \((-2)^2 = 4\).
• Simplifying expressions: After applying substitution and calculating absolute values, the expression \(a^2 - |bc|\) is simplified step-by-step to arrive at the final answer \(4 - 12 = -8\).

By mastering basic algebra, students can comfortably evaluate and manipulate diverse mathematical expressions, which is a vital skill for further mathematical exploration.