Problem 45 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. $$(2-5)(3-6)$$

Step-by-Step Solution

Answer

The expression simplifies to 9.

Step 1: Evaluate Parentheses

Begin by evaluating the expressions inside the parentheses. For the first set of parentheses, calculate \(2 - 5\), which equals \(-3\). For the second set of parentheses, calculate \(3 - 6\), which equals \(-3\). Now the expression simplifies to \((-3)(-3)\).

Step 2: Multiply the Results

Multiply the results of the two parentheses. Calculate \((-3) \times (-3) = 9\). Remember that the product of two negative numbers is a positive number.

Key Concepts

Multiplication Rules

In mathematics, multiplication involves combining equal groups to find their total quantity. When multiplying negative numbers, certain rules apply that simplify our calculations. Here are some key rules to remember:

• Multiplying two positive numbers always yields a positive result. For example, \(2 \times 3 = 6\).
• Multiplying a positive and a negative number results in a negative product, such as \(2 \times (-3) = -6\).
• Multiplying two negative numbers results in a positive product. For instance, \((-3) \times (-3) = 9\), which is why we find in our example that the product is 9 after multiplying \((-3)\) and \((-3)\).

Understanding these rules is crucial for simplifying expressions correctly and ensures that each step in an equation is completed accurately.
Knowing this helps prevent any signs errors when working with expressions involving multiplication.

Addition and Subtraction

Addition and subtraction are foundational arithmetic operations. In any expression, it's important to handle these operations correctly, especially when they are within parentheses. The rules are straightforward:

• Addition combines numbers into a larger value. For example, \(3 + 5 = 8\).
• Subtraction determines the difference between numbers, yielding a smaller value. An example is \(5 - 3 = 2\).
• When subtracting a larger number from a smaller number, the result is negative. Thus, \(2 - 5 = -3\), as seen in the problem's parentheses.

When dealing with operations inside parentheses, solve them first before moving on to other operations. This forms part of the order of operations (PEMDAS/BODMAS), where Parentheses come first followed by Exponents, Multiplication and Division, and finally Addition and Subtraction.
Getting comfortable with these rules will greatly enhance your ability to distinguish what operations need to be tackled first in any mathematical expression.

Simplifying Expressions

Simplifying expressions is about making an expression simpler without altering its value. This process often involves reducing the expression to its simplest form. Here are some general steps:

• Start by resolving any operations within parentheses.
• Follow up with multiplication or division from left to right.
• Finally address any addition or subtraction from left to right.

Applying these steps ensures expression simplification follows the order of operations, thereby rendering accurate results. In the example, solving the operations in parentheses first changed \((2-5)(3-6)\) into \((-3)(-3)\).
Then, multiplying these results simplifies the expression to \(9\). Consistently using these steps will make simplifying expressions feel second nature and improve overall problem-solving skills.