Problem 18 - Intermediate Algebra

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$ [63+(-87)]+(-64) $$

Step-by-Step Solution

Answer

The expression simplifies to -88.

Step 1: Solve Inside the Brackets

Begin by solving the expression inside the square brackets. You have \[63 + (-87)\].This can also be thought of as \[63 - 87\].Calculate:\[63 - 87 = -24\].

Step 2: Combine with the Outside Term

Now, take the result from Step 1 and combine it with the remaining term outside the brackets. You need to solve:\[(-24) + (-64)\].This is the same as:\[-24 - 64\].Calculate:\[-24 - 64 = -88\].

Key Concepts

Numerical Expressions

Numerical expressions are mathematical phrases that use numbers and operations. They can include addition, subtraction, multiplication, and division. Unlike equations, they do not have an equal sign. In order to make sense of these expressions, we need to understand the order of operations, which tells us in what sequence to perform these operations.
For instance, in expressions like

• \( [63 + (-87)] + (-64) \)

, the operations inside the brackets must be done first. Solving numerical expressions accurately is essential because it forms the basis for solving more complex mathematical problems. The goal is to simplify the expression while following the correct order of operations.

Properties of Numbers

Properties of numbers play a crucial role in simplifying mathematical expressions. They act as rules that help us perform operations more efficiently. Some of the key properties include:

• The Commutative Property of addition and multiplication, which states that changing the order of numbers does not change the sum or product, such as a + b = b + a
• The Associative Property, which tells us that the way numbers are grouped does not affect the sum or product, such asa + (b + c) = (a + b) + c
• The Distributive Property, which allows us to multiply a number by a group of numbers added together, such as:a(b + c) = ab + ac

When simplifying \([63 + (-87)] + (-64)\), properties like the associative property can be used to group numbers in ways that make calculations easier.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. It means carrying out all possible operations in the correct order. Here, various rules and properties are applied to eliminate any unnecessary numbers or operations.
In the expression \([63 + (-87)] + (-64)\), we start by performing the operations inside the brackets due to the order of operations: \(63 + (-87)\) simplifies to \(-24\). Next, we combine this result with the remaining numbers: \(-24 + (-64)\), which simplifies further to \(-88\).
The goal is to make the expression as straightforward as possible without changing its value. Properly simplifying expressions helps in understanding the structure and solution of mathematical problems more clearly.