Copyright © Cengage Learning. All rights reserved. 1 Basic Concepts Functions Copyright © Cengage Learning. All rights reserved.

Review of Operations with Whole Numbers Unit 1A Review of Operations with Whole Numbers Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 1.2 Order of Operations Copyright © Cengage Learning. All rights reserved.

Order of Operations The expression 53 means to use 5 as a factor 3 times. We say that 53 is the third power of 5, where 5 is called the base and 3 is called the exponent. Here, 53 means 5  5  5 = 125. The expression 24 means that 2 is used as a factor 4 times; that is, 24 = 2  2  2  2 = 16. Here, 24 is the fourth power of 2.

Order of Operations Just as we use periods, commas, and other punctuation marks to help make sentences more readable, we use grouping symbols in mathematics, such as parentheses “( )” and brackets “[ ],” to help clarify the meaning of mathematical expressions. Parentheses not only give an expression a particular meaning, they also specify the order to be followed in evaluating and simplifying expressions.

Order of Operations What is the value of 8 – 3  2? Is it 10? Is it 2? Or is it some other number? It is very important that each mathematical expression have only one value. For this to happen, we all must not only perform the exact same operations in a given mathematical expression or problem but also perform them in exactly the same order.

Order of Operations The following order of operations is followed by all. Order of Operations 1. Always do the operations within parentheses or other grouping symbols first. 2. Then evaluate each power, if any. Examples: 4  32 = 4  (3  3) = 4  9 = 36 52  6 = (5  5)  6 = 25  6 = 150

Order of Operations 3. Next, perform multiplications and divisions in the order in which they appear as you read from left to right. For example, 60  5  4  3  2 = 300  4  3  2 = 75  3  2 = 25  2 = 50

Order of Operations 4. Finally, perform additions and subtractions in the order in which they appear as you read from left to right. Note: If two parentheses or a number and a parenthesis occur next to one another without any sign between them, multiplication is indicated. By using the above procedure, we find that 8 – 3  2 = 8 – 6 = 2.

Example 1 Evaluate: 2 + 5(7 + 6). = 2 + 5(13) = 2 + 65 = 67 Note: A number next to parentheses indicates multiplication. In Example 1, 5(13) means 5  13. Adjacent parentheses also indicate multiplication: (5)(13) also means 5  13. Add within parentheses. Multiply. Add.

Order of Operations If pairs of parentheses are nested (parentheses within parentheses, or within brackets), work from the innermost pair of parentheses to the outermost pair. That is, remove the innermost parentheses first, remove the next innermost parentheses second, and so on.