Problem 37 - Beginning and Intermediate Algebra

Question

Find each sum. $$ 6+[12+(-3)] $$

Step-by-Step Solution

Answer

15

Step 1: Evaluate the innermost parentheses

First, solve the expression inside the innermost parentheses: \there, which is \( 6 + (12 + (-3)) \). Inside the parentheses, add 12 and -3 first: \( 12 + (-3) = 12 - 3 = 9 \).

Step 2: Simplify the expression

Now that the inner parentheses have been evaluated, the expression simplifies to: \( 6 + 9 \).

Step 3: Perform the final addition

Finally, add the remaining numbers: \( 6 + 9 = 15 \).

Key Concepts

Understanding Arithmetic Operations

Arithmetic operations are basic mathematical tasks that we perform every day. These include addition, subtraction, multiplication, and division. In solving the given exercise, we primarily focus on addition, including the combination of positive and negative integers. Understanding these operations helps us solve problems effortlessly and ensures accuracy in computations. In the exercise provided, the first operation involves adding \(12 + (-3)\). Here, we need to understand that adding a negative number is the same as subtracting the number. This simplifies our task and keeps our calculations straightforward.

Mastering Order of Operations

The order of operations is a set of guidelines that tells us the sequence in which operations should be performed in a mathematical expression. This is often remembered with the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition, and Subtraction (from left to right).
In our current problem, we see parentheses which need to be dealt with first before any other operation. Performing operations inside parentheses ensures that we tackle the innermost expressions first. This is why we start with \(12 + (-3)\) before adding the result to 6.

The Simplicity of Integer Addition

Integer addition involves combining whole numbers, which could be positive or negative. It's crucial to know the rules around adding these integers. Here are a few basic rules that can help: 1. If both numbers are positive, simply add them. 2. If both numbers are negative, add their absolute values and apply a negative sign. 3. If one number is negative and the other positive, subtract the smaller absolute value from the larger.
In the exercise, we combined positive and negative integers in the parentheses operation \(12 + (-3)\). Simplifying this to 9, allowed us to add it to the previous integer 6 seamlessly.