Problem 44 - 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. $$(9-3)(2-6)$$

Step-by-Step Solution

Answer

The simplified expression is \(-24\).

Step 1: Evaluate Parentheses

Start by evaluating the operations within the parentheses. For the first set of parentheses, we have \(9 - 3\). Calculate this to get \(6\). For the second set of parentheses, we have \(2 - 6\), which simplifies to \(-4\). After evaluating the parentheses, the expression becomes \((6)(-4)\).

Step 2: Apply Multiplication

The expression has been simplified to two numbers to be multiplied: \( (6)(-4) \). Multiplying these together gives \( 6 \times -4 = -24 \). Remember, a positive number times a negative number results in a negative product.

Key Concepts

Simplifying Expressions

Simplifying expressions involves performing operations to make an expression as easy to understand as possible. The goal is to reduce the complexity and work out operations in a proper sequential manner. People often use the acronym PEMDAS to remember the order of operations:

• Parentheses
• Exponents (though not used here)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Faced with an expression like \((9-3)(2-6)\), the first step in simplifying is to carefully address and solve the operations within each set of parentheses. Then, move on to the next operation which, in this case, is multiplication.

Multiplication

Multiplication is a core mathematical operation where you combine groups of equal sizes. It is one of the quickest ways to add groups of numbers. For example, multiplying 6 by -4 can be thought of as taking away 4 six times, resulting in \[-24\]. A key rule to remember in multiplication is the sign rule:

• Positive times positive equals positive
• Positive times negative equals negative
• Negative times positive equals negative
• Negative times negative equals positive

In the case of \((6)(-4)\), the positive number 6 is multiplied by the negative number -4, yielding a negative product: \(-24\). This principle helps in predicting the sign of the product without needing to carry out extensive calculations each time.

Parentheses

Parentheses greatly influence the outcome of expressions by signaling which calculations should be completed first. Consider them your mathematical way of setting priorities. In \((9-3)(2-6)\), we are directed to first solve the operations within the parentheses:

• First \(9 - 3\), which simplifies to 6
• Then, \(2 - 6\), which simplifies to -4

This simplifies the expression so the operation outside the parentheses can be addressed without error. Once you process inside the parentheses, you move onto multiplication or any other operation outside them. This keeps calculations clear and accurate, reducing errors in more complex math problems.