Problem 32 - 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. $$7(-6+3)$$

Step-by-Step Solution

Answer

The simplified expression is -21.

Step 1: Identify Operations Inside Parentheses

The expression given is \(7(-6+3)\). According to the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction), operations inside the parentheses \((-6+3)\) should be simplified first.

Step 2: Simplify Inside Parentheses

Calculate \(-6 + 3\). Subtract 3 from 6, which results in \(-3\). Now the expression simplifies to \(7(-3)\).

Step 3: Multiply

With the simplified expression \(7(-3)\), multiply 7 by \(-3\). This results in \(-21\).

Key Concepts

Parentheses in Mathematics

In math, parentheses are extremely important as they determine the order in which operations are performed. Parentheses tell you to perform the operations enclosed within them first, before moving on to other calculations. This is a part of the broader guideline known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

For example, in the expression \(7(-6+3)\), the parentheses around \(-6+3\) signal to perform that calculation first. So, you don’t move on to multiplication or any other operation until you have simplified what’s inside the parentheses.

It’s like organizing your tasks. Deal with what’s inside your parentheses right away, before you get to the other stuff in your to-do list. Without respecting this order, answers could differ, as other operations might be wrongly prioritized.

Multiplication Rules

Multiplication, an arithmetic operation, requires two numbers, known as factors, to be combined into a product. However, it's worth noting a couple of rules that might make you a multiplication master!

When dealing with multiplication, always ensure that you follow the proper sequence. As indicated by PEMDAS, multiplication comes after you've handled any operations in parentheses. So after simplifying \((-6+3)\) to \(-3\), in our example \(7(-3)\), you carry out the multiplication \(7 \times -3\).

Additionally, note the rules when multiplying negative numbers:

• A positive times a positive results in a positive number.
• A positive times a negative results in a negative number.
• A negative times a negative results in a positive number.

In the expression \(7(-3)\), since 7 is positive and \(-3\) is negative, the product is negative \(-21\). Remembering these rules can help you avoid mistakes in calculations involving multiplication.

Negative Numbers

Understanding negative numbers can change how you view number-related problems! Negative numbers are typically represented with a minus sign in front of them. They allow mathematicians to extend the number line in the opposite direction of positive numbers.

They are crucial in real-life applications such as temperature below zero or finances in debt. In the expression \(7(-6+3)\), you're dealing with negatives and need to find their sum or product.

Here are some basic rules to handle negative numbers:

• If you add a negative number, it's like subtracting the positive version of that number (e.g., \(-6 + 3\) is the same as \(3 - 6\)).
• Subtracting a negative number is like adding its positive counterpart.
• Multiplying by a negative number reverses the sign of the other factor.

So, when simplifying \(-6 + 3\) in our expression, you essentially subtract 6 from 3, yielding \(-3\). Later, multiplying this \(-3\) by 7 flips the product's sign, giving you \(-21\). Embracing these rules will make negative numbers a lot less intimidating.