Problem 17 - Algebra for College Students

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$[83+(-99)]+18$$

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Simplify Inside the Brackets

The expression given is \(83 + (-99)\) + 18. Use the properties of addition to simplify the expression inside the brackets first. Add the numbers: \(83 + (-99) = 83 - 99 = -16\).

Step 2: Add Remaining Number

Now, add the result from Step 1 to the outside number: \(-16 + 18\). Using the properties of addition, simplify this to \-16 + 18 = 2\.

Key Concepts

Properties of Addition

The properties of addition are fundamental to simplifying any expression effectively. These properties help us to rearrange and group numbers in a way that makes calculations simpler. Here we discuss the most relevant ones.

• Commutative Property: This property states that the order of numbers does not affect their sum. In other words, 2 + 3 is the same as 3 + 2.
• Associative Property: This property tells us that when adding three or more numbers, the way in which the numbers are grouped does not affect the sum. For example, (1 + 2) + 3 is the same as 1 + (2 + 3).
• Identity Property: Adding zero to any number will not change its value. For instance, 7 + 0 equals 7.

In our original exercise, we used the commutative and associative properties to simplify the expression within the brackets before tackling the entire expression. This step-by-step approach helps us deal with complex expressions in a manageable way.

Negative Numbers

Understanding negative numbers is crucial when working with numerical expressions that involve subtraction or addition of signed numbers. A negative number is one that is less than zero and is usually denoted by a minus sign (-).
When adding negative numbers to positive numbers, it's like taking steps forward and then steps back. To simplify, you subtract the smaller from the larger and keep the sign of the larger number. For example, in the original exercise, we have 83 + (-99). Here:

• Think of 83 as steps forward from zero.
• Think of (-99) as steps backward.
• Calculate it as 83 - 99, giving you -16, as you have more steps backward.

Handling negative numbers may seem tricky at first, but with practice, it's straightforward. Always remember to pay attention to the signs!

Numerical Expressions

Numerical expressions are combinations of numbers and operations, such as addition or subtraction, without an equality or inequality sign. Simplifying numerical expressions usually requires careful application of arithmetic operations and properties of numbers.
Here's a simple guide on how to tackle them properly:

• Identify the numbers and operations: Look carefully at the expression to understand what operations are being used.
• Apply order of operations: Use the proper sequence of operations (brackets, exponents, multiplication and division, addition, and subtraction) to simplify correctly.
• Simplify step-by-step: Break down the expression into simpler parts, starting with operations inside parentheses.

In our specific problem, we simplified the numerical expression \(83 + (-99) + 18\) by first focusing inside the brackets, then dealing with the rest in a methodical way. Clear, step-by-step simplification ensures accuracy in solving any problem related to numerical expressions.