Problem 53 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. $$(-10+4)+(-3+12)$$

Step-by-Step Solution

Answer

The simplified expression is 3.

Step 1: Simplify Inside Parentheses

Start by simplifying the expressions inside the parentheses. The first expression is \(-10 + 4\) and the second one is \(-3 + 12\). Calculate each: \(-10 + 4 = -6\) and \(-3 + 12 = 9\).

Step 2: Perform Addition

With the simplified expressions from Step 1, add \(-6 + 9\).Calculate this: \(-6 + 9 = 3\).

Key Concepts

Parentheses in Order of Operations

When working with mathematical expressions, parentheses indicate which part of the expression you should calculate first. This is a fundamental rule in the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). In our exercise, we have the expression \((-10+4)+(-3+12)\). The parentheses around \(-10+4\) and \(-3+12\) tell us to deal with these expressions first before adding them together. Focusing on these parts keeps calculations organized and accurate. You must always resolve expressions inside parentheses first, even if they are nested or layered inside other parentheses. Key Takeaway: Always start simplifying expressions by addressing parentheses first, as they determine the first calculations you perform.

Addition in Mathematics

Addition is one of the basic operations in math, symbolized by the plus sign \(+\). It is used to combine values or increase quantities. In our exercise, we first simplified the expressions inside the parentheses and then used addition to combine these results. After simplifying \(-10+4\) to \(-6\) and \(-3+12\) to \(9\), the last step is to add these two results together: \(-6 + 9\). This operation involves finding the difference between the two numbers since they have opposite signs. When you add positive and negative numbers, as in this example, you essentially subtract their absolute values and keep the sign of the larger number.Key Points to Remember:

• Adding two positive numbers gives a positive result.
• Adding two negative numbers gives a negative result.
• Adding a positive and a negative number involves subtraction, and the result takes the sign of the number with the greater absolute value.

Simplifying Expressions

Simplifying expressions involves making them easier to understand and work with, by performing basic arithmetic operations and reducing the expressions to their simplest form. In the given expression \(-10+4\) and \(-3+12\), each part inside the parentheses was simplified first to make the whole expression more manageable. By dealing with these straightforward calculations first, it becomes easier to perform the final addition operation. Why Simplify Expressions?

• It reduces complexity, making it easier for you to solve or further manipulate expressions.
• It helps avoid mistakes by breaking down complex problems into simple steps.
• It often reveals equivalent expressions or solutions, thus enhancing understanding and problem-solving skills.

Simplifying is a crucial step and sometimes even the simplest expressions, like \(-6 + 9\), benefit from being broken down in this manner to better grasp the process and outcome.