Problem 53 - 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. $$3(-2)(6-7)$$

Step-by-Step Solution

Answer

The expression simplifies to 6.

Step 1: Simplify inside the parentheses

First, look at the expression within the parentheses, which is \(6 - 7\). Perform the subtraction inside the parentheses to simplify it. \(6 - 7 = -1\). Substitute \(-1\) back into the expression, resulting in \(3(-2)(-1)\).

Step 2: Multiply the numbers together

Now that the expression is simplified to \(3(-2)(-1)\), proceed with multiplication. You can multiply these numbers in any order due to the commutative property of multiplication. \((-2)\times(-1) = 2\) because the product of two negative numbers is positive. Then \(3\times 2 = 6\).

Key Concepts

Understanding Addition

Addition is one of the most fundamental operations in mathematics. It's the process of combining two or more numbers to get a total or sum. Imagine having three apples and adding two more, you'll end up with five apples. This is addition at work. With numbers, when you add, you're essentially increasing the count or value. An important thing to remember about addition is its commutative property. This means that the order of numbers doesn't matter. For example:

• 3 + 2 = 5
• 2 + 3 = 5

No matter how you arrange the numbers, the result is the same, and this makes calculations flexible and easier to manage. Addition is often paired with subtraction since they're inverse operations.

Grasping Subtraction

Subtraction, as the opposite of addition, involves finding the difference between numbers. Simply put, it's removing a number from another. If you start with five apples and take away three, you're left with two apples. When you subtract, you're typically decreasing the total. A key feature of subtraction is that it's not commutative. The order in which you subtract does matter. For example:

• 5 - 2 = 3
• 2 - 5 = -3

As you can see, switching the order gives you a different result. In expressions with multiple operations, subtraction often follows the rules for order of operations: parentheses, exponents, multiplication and division, and finally, addition and subtraction.In the expression from the original exercise, the subtraction inside the parentheses was the first thing to solve: \(6 - 7 = -1\). This step is crucial to having the right numbers to multiply afterwards.

Mastering Multiplication

Multiplication can be thought of as repeated addition. If you have four groups of three apples, instead of adding three four times (\(3 + 3 + 3 + 3\)), you can simply multiply: \(4 \times 3 = 12\). Multiplication is incredibly useful for quickly scaling numbers and performing more complex calculations efficiently.An important property of multiplication is that it's commutative, like addition. This means you can multiply numbers in any order and get the same result, which adds flexibility:

• 3 \times 2 = 6
• 2 \times 3 = 6

Another interesting property is the multiplication of negatives. When two negative numbers are multiplied, the result is positive, as seen in the exercise example where \((-2) \times (-1)\) became 2. After simplifying the parentheses and handling subtraction, multiplication helps reach the final expression value, connecting all the operations seamlessly. In this way, multiplication plays a pivotal role in solving and simplifying expressions.