Problem 40 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$(-2)^{2}-3(-2)(6)-(-5)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is 15.

Step 1: Simplify (-2)²

Calculate the square of (-2). Recall that squaring a negative number involves multiplying the number by itself. \[ (-2)^2 = (-2) imes (-2) = 4 \]

Step 2: Simplify -3(-2)(6)

Multiply -3, (-2), and 6 together. Start with multiplying -3 and -2.\[ -3 imes (-2) = 6 \] Then, multiply the result by 6. \[ 6 imes 6 = 36 \] So, -3(-2)(6) = 36.

Step 3: Simplify - (-5)²

Calculate the square of (-5) and negate the result. \[ (-5)^2 = (-5) imes (-5) = 25 \] Then apply the negative sign: \[ -(-5)^2 = -25 \]

Step 4: Combine the Results

Now, combine the results from Steps 1, 2, and 3 into a single expression. \[ 4 + 36 - 25 \] Perform the addition and subtraction from left to right. First, add 4 and 36:\[ 4 + 36 = 40 \] Then, subtract 25 from 40: \[ 40 - 25 = 15 \]

Key Concepts

Squaring Numbers

When we talk about squaring numbers, we're referring to multiplying a number by itself. This operation is applicable to both positive and negative numbers. Understanding the difference is crucial.

• For any positive number, squaring results in a larger positive number because you multiply two positives together.
• With negative numbers, squaring creates a positive result as well—in our exercise, squaring (-2) gave us 4. This happens because a negative times a negative equals a positive.

Hence, pay close attention to parentheses in expressions, like (-2)^2, indicating the entire negative number is squared. Mistakes here can completely change your answer.

Order of Operations

Math is all about getting the steps in the right order. The order of operations is a set of rules that tells us the order in which to solve parts of an expression. This is often remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (like squaring numbers)
• M and D: Multiplication and Division (from left to right)
• A and S: Addition and Subtraction (from left to right)

In our exercise, we apply PEMDAS when simplifying (-2)^2 - 3(-2)(6) - (-5)^2. Start with the parentheses and exponents, followed by multiplication, and finally addition and subtraction.

Negative Numbers

Negative numbers can sometimes be tricky, especially when they interact with operations like squaring or multiplying. Some easy rules can help:

• When multiplying or dividing two negative numbers, the result is positive.
• Squaring a negative number also results in a positive number, as seen with (-5)^2, which converts to 25.
• However, a minus sign directly in front of such terms can change the story; for example, -(-5)^2 means you take that positive result and apply a negative sign again, turning it into -25.

Always keep track of these rules, especially when negative signs are involved in multiple steps.