5.4 Order of Operations with Real Numbers
Learning Objective(s) 1 Use the order of operations to simplify expressions. 2 Simplify expressions containing absolute values.
Introduction
Recall the order of operations discussed previously.
The Order of Operations
• Perform all operations within grouping symbols first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and fraction bars. • Evaluate exponents or square roots. • Multiply or divide, from left to right. • Add or subtract, from left to right.
This order of operations is true for all real numbers.
Example Problem Simplify 7 – 5 + 3 • 8. 7 – 5 + 3 • 8 According to the order of operations, multiplication comes before addition and subtraction. Multiply 3 • 8. 7 – 5 + 24 Now, add and subtract from left to right. 7 – 5 comes first. 2 + 24 = 26 Finally, add 2 + 24. Answer 7 – 5 + 3 • 8 = 26
When you are applying the order of operations to expressions that contain fractions, decimals, and negative numbers, you will need to recall how to do these computations as well.
5.43 Example Problem 11 Simplify 3•−÷ 8 34 11According to the order of 3•−÷ 8 34operations, multiplication comes before addition and subtraction. 1 Multiply 3• first. 3 1 1 18−÷ Now, divide 8 ÷ . 4 4
1 84 8÷= • = 32 4 11 1 – 32 = −31 Subtract. Answer 11 3 •−÷ 8 =− 31 34
Exponents
When you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as 72 is exponential notation for 7 • 7. (Exponential notation has two parts: the base and the exponent or the power. In 72 , 7 is the base and 2 is the exponent; the exponent determines how many times the base is multiplied by itself.)
Exponents are a way to represent repeated multiplication; the order of operations places it before any other multiplication, division, subtraction, and addition is performed.
Example
Problem Simplify 323 •2 .
323 •2 This problem has exponents and multiplication in it. According to the order of operations, simplifying 32 and 23 comes before multiplication.
9•23 32 is 3 • 3, which equals 9.
9•8 23 is 2 • 2 • 2, which equals 8. 9 • 8= 72 Multiply.
Answer 323 • 2= 72
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Example
23 Problem 11 Simplify + 32 . 24
23 11 This problem has exponents, + 32 multiplication, and addition in it. 24 According to the order of operations, simplify the terms with the exponents first, then multiply, then add.
3 2 11 1 11 1 + 32 Evaluate: = = 44 2 22 4 11 3 + 32 1 111 1 Evaluate: = = 4 64 4 444 64 1 32 + Multiply. 4 64 113 32÷ 32 1 += Simplify. = , so you can 424 64÷ 32 2 11 add + . 42
23 Answer 11 3 += 32 24 4
Self Check A Simplify: 100− 52 4 .
When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first.
Remember that parentheses can also be used to show multiplication. In the example that follows, both uses of parentheses—as a way to represent a group, as well as a way to express multiplication—are shown.
5.45 Example Problem Simplify (3 + 4)2 + (8)(4). (3 + 4)2 + (8)(4) This problem has parentheses, exponents, multiplication, and addition in it. The first set of parentheses is a grouping symbol. The second set indicates multiplication.
(3 + 4)2 + (8)(4) Grouping symbols are handled first. Add numbers in parentheses. 72 + (8)(4) Simplify 72. 49 + (8)(4) Perform multiplication. 49 + 32 = 81 Perform addition. Answer (3 + 4)2 + (8)(4) = 81
Example Problem Simplify (1.5 + 3.5) – 2(0.5 • 6)2.
(1.5 + 3.5) – 2(0.5 • 6)2 This problem has parentheses, exponents, multiplication, subtraction, and addition in it.
Grouping symbols are handled first. Add numbers in the first set of parentheses. 5 – 2(0.5 • 6)2 Multiply numbers in the second set of parentheses. 5 – 2(3)2 Evaluate exponents.
5 – 2 • 9 Multiply.
5 – 18 = −13 Subtract.
Answer (1.5 + 3.5) – 2(0.5 • 6)2 = −13
5.46 Example Problem 5−+ [3 (2 • ( − 6))] Simplify 322 + 5−+ [3 (2 • ( − 6))] This problem has brackets, parentheses,
322 + fractions, exponents, multiplication, subtraction, and addition in it.
Grouping symbols are handled first. The parentheses around the -6 aren’t a grouping symbol, they are simply making it clear that the negative sign belongs to the 6. Start with the innermost set of parentheses that are a grouping symbol , here it is in the numerator of the fraction, (2 • −6), and begin working out. (The fraction line acts as a type of grouping symbol, too; you simplify the numerator and denominator independently, and then divide the numerator by the denominator at the end.) 5− [3 +− ( 12)] Add the values in the brackets.
322 + 5−− [ 9] Subtract 5 – [−9] = 5 + 9 = 14.
322 + 14 The top of the fraction is all set, but the
322 + bottom (denominator) has remained untouched. Apply the order of operations to that as well. Begin by evaluating 32 = 9. 14
92+ Now add. 9 + 2 = 11. 14
11
Answer 5−+ [3 (2 • ( − 6))] 14 = 32 + 2 11
Self Check B 2 333 + Simplify +1. (2)(3)−−
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Absolute Value Expressions Objective 2
Absolute value expressions are one final method of grouping that you may see. Recall that the absolute value of a quantity is always positive or 0.
When you see an absolute value expression included within a larger expression, follow the regular order of operations and evaluate the expression within the absolute value sign. Then take the absolute value of that expression. The example below shows how this is done.
Example Problem 3+ 2− 6 Simplify 2 3 1.5−− ( 3)
3+− 26 This problem has absolute values, decimals, multiplication, subtraction, and addition in it. 2 3 1.5−− ( 3) Grouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator. Evaluate |2 – 6|. 34+− Take the absolute value of |−4|.
2 3 1.5−− ( 3)
34+ Add the numbers in the numerator.
2 3 1.5−− ( 3)
7 Now that the numerator is simplified, turn to
2 3 1.5−− ( 3) the denominator. Evaluate the absolute value expression first. 7 The expression “2|4.5|” reads “2 times the
2 4.5−− ( 3) absolute value of 4.5.” Multiply 2 times 4.5.
7
2 4.5−− ( 3) 7 Subtract.
9−− ( 3) 7
12
Answer 3+− 26 7 = 2 3 1.5−− ( 3) 12
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Self Check C Simplify: (5|3 – 4|)3.
5.4 Self Check Solutions
Self Check A Simplify: 100− 52 4 .
To simplify this expression, simplify the term with the exponent first, then multiply, then subtract. 52 = 25, and 25 • 4 = 100, and 100 – 100 = 0.
Self Check B 2 333 + Simplify +1. (2)(3)−−
The entire quantity within the brackets is 5. 52 is 25, and 25 + 1 = 26.
Self Check C Simplify: (5|3 – 4|)3.
|3 – 4| = |−1| = 1, and 5 times 1 is 5. 5 cubed is 125.
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