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Chapter 1: Problem 43

Rewrite each expression using the indicated property. \((-5+3)+6 ;\) associative

Short Answer

Expert verified
Using the associative property, we can rewrite the expression \((-5+3)+6\) as \((-5+(3+6))\). When simplified, this evaluates to 4.

Step by step solution

01

Identify the numbers and grouping

In the expression \((-5+3)+6\), there are three numbers being added together: 1. -5 2. 3 3. 6 Currently, the numbers -5 and 3 are grouped together.
02

Apply the associative property

To apply the associative property, we can simply change the grouping of the numbers being added. Instead of adding -5 and 3 first, we can add 3 and 6 first, then add -5 to their sum. The altered grouping will look like this: \((-5+(3+6))\)
03

Simplify the expression

Now, let's simplify the expression. First, add the numbers inside the parentheses: \((-5+(3+6)) = (-5+9)\) Now, add -5 and 9: \((-5+9) = 4\) So, \((-5+3)+6\) = \((-5+(3+6))\) = 4

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Addition
Addition is a basic mathematical operation that involves combining numbers to form a total. It is one of the four fundamental operations of arithmetic. When you add numbers together, like in the expression \( -5 + 3 + 6 \) , you are looking to find the total sum of these values.
Addition is commutative, which means you can change the order of the numbers without affecting the result.
  • Example: \( 3 + 4 = 4 + 3 \)
This operation is essential in grouping and rearranging numbers, particularly when using properties like the associative property to simplify expressions.
Understanding addition helps in managing and simplifying expressions correctly, providing the foundation needed for more complex calculations.
Grouping Symbols
Grouping symbols like parentheses \( \,\left(\right) \) are used to dictate the order of operations in arithmetic expressions. They help identify which operations should be performed first. In the expression \((-5+3)+6\), the parentheses indicate that -5 and 3 should be added together before any other operations.
  • Example: \( (2 + 3) + 4 \) - Do the \( 2+3 \) first.
These symbols are crucial in expressions to override default operation order, which generally follows the "PEMDAS" rule (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).Proper use of grouping symbols can simplify the computation process, making expressions easier to manage and solve.
Arithmetic Expressions
Arithmetic expressions consist of numbers and operations like addition, subtraction, multiplication, and division. They represent mathematical quantities and are often used to pose questions or present problems in a clearer way.
For instance, \( -5 + 3 + 6 \) is an arithmetic expression where the task is to calculate the sum of all included numbers.Expressions can become complex, involving multiple operations and layers of grouping symbols. It鈥檚 important to understand how to navigate these different components effectively to find the solution.
  • Focus on operations inside grouping symbols first.
  • Be mindful of the order in which you perform operations to avoid errors.
Becoming proficient in reading and solving arithmetic expressions is crucial for tackling more advanced mathematical problems.
Simplification Process
The simplification process involves altering expressions into a simpler or more manageable form without changing their value. Using properties like the associative property can aid in this process.
For example, starting from the expression \((-5 + 3) + 6\), we change how the terms are grouped: \((-5 + (3 + 6))\).By addressing operations within grouping symbols first, we simplify:
  • Calculate \(3 + 6 = 9\)
  • Then add \(-5 + 9 = 4\)
Thus achieving the simplified form. This process not only requires understanding the properties of numbers and operations but also enhances problem-solving skills to solve efficiently.
Simplification aids in reducing errors and provides a neat answer to complex arithmetic problems.

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Most popular questions from this chapter

Objectives 3 and 4 A welder must construct a beam with a total length of \(32 \frac{7}{8}\) in. If he has already joined a \(14 \frac{1}{6}\) -in. beam with a \(10 \frac{3}{4}-i n .\) beam, find the length of a third beam needed to reach the total length.

Use the order of operations to simplify. $$-2^{3}-(-3)(4)+5[(-9+30) \div 7]$$

Write a mathematical expression for each and simplify. 19 less than the product of \(-4\) and \(-8\)

Determine whether each statement is true or false. For any real numbers \(a\) and \(b\). $$|a-b|=|b-a|$$

Use the order of operations to simplify. $$-23-6^{2} \div 4$$
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