Problem 41 - Elementary and Intermediate Algebra

Question

Simplify. $$ (8-2)(3-9) $$

Step-by-Step Solution

Answer

-36

Step 1 - Calculate Inside the Parenthesis

First, solve the expressions inside each parenthesis: For the first parenthesis: \(8 - 2 = 6\) For the second parenthesis: \(3 - 9 = -6\)

Step 2 - Multiply the Results

Next, multiply the results of the two calculations: \(6\) and \(-6\) \(6 \times (-6) = -36\)

Key Concepts

Parentheses in Math

When you see parentheses in a math problem, it means you need to perform the calculations inside them first. Parentheses help to group parts of an expression together, indicating that the enclosed operations should be done before dealing with the rest of the expression. For instance, in the problem \( (8 - 2)(3 - 9) \), you need to solve \( 8 - 2 \) and \( 3 - 9 \) before tackling anything else.
This grouping ensures that everyone solves the problem the same way. Another example: in \( (2 + 3) \times 4 \), you do \( 2 + 3 \) first to get 5, then multiply by 4 to get 20.

• Solve inside the parentheses: this is the first step.
• Groups and orders parts of an expression.

Remember: always look for parentheses first in any expression!

Order of Operations

Understanding the order of operations is key to solving math expressions correctly. The order of operations specifies the correct sequence in which to solve parts of a math problem. This is often remembered by the acronym PEMDAS:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

Parentheses come first, which is why in our problem \( (8-2)(3-9) \) we start by solving inside the parentheses. After resolving the expressions inside the parentheses, multiplication comes next.
If you have a math problem with different operations inside and outside parentheses, always follow PEMDAS. For instance, in the problem \( (2+5)^2 - (3 \times 4) \), you would:

• Solve inside the parentheses: \(7^2 - 12\)
• Calculate exponents: \(49 - 12\)
• Finish with subtraction: \(37\)

Following the order of operations ensures that your solutions are accurate and consistent.

Multiplication of Integers

Multiplying integers can sometimes be tricky, especially when dealing with negative numbers. A positive times a positive, and a negative times a negative, will always result in a positive. However, a negative times a positive will always be negative. This rule is crucial when simplifying expressions. In our solution:
1. After solving inside the parentheses, we get \(6 \times (-6) \).
2. Following the rules: a positive times a negative gives us a negative result.
3. Thus, \(6 \times (-6) = -36\).

• Positive \times Positive = Positive (e.g., \( 3 \times 4 = 12 \))
• Negative \times Negative = Positive (e.g., \( -3 \times -4 = 12 \))
• Positive \times Negative = Negative (e.g., \( 3 \times -4 = -12 \))

Understanding these rules helps to simplify integers multiplication effectively.