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Chapter 1: Problem 110

In the following exercises, simplify using the order of operations. $$ 3^{2}-18 \div(11-5) $$

Short Answer

Expert verified
6

Step by step solution

01

Evaluate Parentheses

First, solve the expression inside the parentheses (11 - 5).\[11 - 5 = 6\]
02

Division

Next, perform the division 18 divided by the result obtained from the parentheses, which is 6.\[18 \div 6 = 3\]
03

Exponentiation

Evaluate the exponent next, which is \(3^2\):\[3^{2} = 9\]
04

Subtraction

Finally, subtract the result of the division from the result of the exponentiation:\[9 - 3 = 6\]

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Key Concepts

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

simplification
In mathematics, simplification involves making an expression easier to manage and understand. When encountering mathematical expressions, it's essential to simplify them using the order of operations, also known as PEMDAS (Parentheses, Exponentiation, Multiplication, Division, Addition, Subtraction). This ensures that the calculations are performed in the correct sequence to arrive at the correct result. For our exercise, simplifying the expression means evaluating it step-by-step according to these rules.
parentheses
Parentheses are crucial in math because they indicate which operations should be performed first. In the given expression, we start by solving the part inside the parentheses:
  • Expression: \( 3^{2}-18 \div(11-5) \)
  • Step: Solve \(11-5 \)
  • Result: \(6\)
By solving the parentheses first, we ensure that the rest of the calculations can proceed accurately.
exponentiation
Exponentiation is the process of raising a number to a power. The exponent tells us how many times to multiply the base by itself. In our exercise, we encounter the exponent \(3^{2} \), which means:
  • Base: 3
  • Exponent: 2
  • Calculation: \(3 \times 3 = 9\)
Evaluating the exponentiation early ensures the rest of the operations align correctly according to the order of operations.
division
Division splits a number into equal parts. In terms of the order of operations, it's essential to perform division before any addition or subtraction unless parentheses indicate otherwise. For our exercise:
  • Expression: \( 18 \div 6 \)
  • Calculation: Divide 18 by the result obtained from the parentheses.
  • Result: \( 3 \)
Performing the division at this step ensures all elements of the expression are accurately processed.
subtraction
Subtraction involves taking one number away from another. It's usually one of the final steps in the order of operations. In our problem, after performing exponentiation and division, we are left with the subtraction:
  • Expression: \( 9 - 3 \)
  • Calculation: Subtract the result of the division from the result of the exponentiation.
  • Result: \( 6 \)
Completing this final step gives us the simplified expression's value.

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