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Chapter 2: Problem 104

Simplify each of the following according to the rule for order of operations. $$6-2(9-11)$$

Short Answer

Expert verified
The simplified result is 10.

Step by step solution

01

Parentheses Evaluation

Start by evaluating the expression inside the parentheses. In this case, inside the parentheses we have \((9 - 11)\). First, subtract \(11\) from \(9\) to get \(-2\). So, the expression simplifies from \(6 - 2(9 - 11)\) to \(6 - 2(-2)\).
02

Multiplication

Next, perform the multiplication operation. Multiply \(-2\) by \(-2\) from the expression \(6 - 2(-2)\). The result is \(4\) because multiplying two negative numbers yields a positive result.
03

Subtraction

Lastly, perform the subtraction operation. Substitute the result from the multiplication to the expression: \(6 + 4\). Add \(4\) to \(6\) which results in \(10\).

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Key Concepts

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

Understanding Parentheses
Parentheses play a crucial role in mathematical expressions, guiding us on which operation to perform first. This is part of the rule known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By following this rule, we solve expressions systematically from left to right.

In the exercise given, we must start by evaluating the expression inside the parentheses:
  • The expression is \(9 - 11\).
  • Inside the parentheses, perform the subtraction: \(9 - 11 = -2\).
Once simplified, the expression becomes \(6 - 2(-2)\). The parentheses ensured that we completed this first operation before proceeding further.
The Role of Multiplication
Next, according to the order of operations, we perform multiplication. In our exercise, after simplifying the parentheses, we have \(6 - 2(-2)\). Here, the multiplier is understood to be directly affecting the result from the parentheses.

Let's walk through the process:
  • Multiply \(2\) by \(-2\) to get \(4\).
  • This is because multiplying a negative number by another negative number results in a positive number.
Now, we've simplified our expression further to \(6 + 4\). Multiplication is always resolved before any addition or subtraction that is not within parentheses.
Executing Subtraction
Finally, we reach the subtraction step. In our current expression, after all previous operations, we simplify \(6 + 4\).

At this stage, subtraction or addition can be solved freely, since all operations within parentheses and multiplication have been completed.
  • Here, you simply add the numbers: \(6 + 4 = 10\).
  • This gives us the final simplified result of the expression.
This step demonstrates how subtraction affects an expression once all previous rules are observed. By adhering to the order of operations, we arrive at the correct solution in a straightforward manner.

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Most popular questions from this chapter

Give an example that shows that subtraction is not a commutative operation.

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-8-4-2$$

In Chapter 1 we defined an arithmetic sequence as a sequence of numbers in which each number, after the first number, is obtained from the previous number by adding the same amount each time. Find the next two numbers in each arithmetic sequence below. $$10,5,0, \dots$$

Perform the indicated operations. $$4^{3}$$

Subtract. $$-79-21$$
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