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Sect 1.7 - Exponents, Square Roots and the Order of Operations

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Sect 1.7 - Exponents, Square Roots and the Order of Operations
Objective a: Understanding and evaluating exponents. We have seen that multiplication is a shortcut for repeated addition and division is a shortcut for repeated subtraction. Exponential notation is a shortcut for repeated multiplication. Consider the follow: 5•5•5•5•5•5 Here, we are multiplying six factors of five. We will call the 5 our base and the 6 the exponent or power. We will rewrite this as 5 to the 6
th
power: 5•5•5•5•5•5 = 5
6
The number that is being multiplied is the base and the number of factors of that number is the power. Let's try some examples: Write the following in exponential form: Ex. 1a 8•8•8•8 Ex. 1b 7•7•7•7•7 Solution: Solution: Since there are four factors Since there are five factors of 8, we write 8
4
. of 7, we write 7
5
. Simplify the following: Ex. 2a 3
4
Ex. 2b 7
3
Solution: Solution: Write 3
4
in expanded form Write 7
3
in expanded form and multiply: and multiply: 3
4
= 3•3•3•3 = 9•3•3 = 27•3 = 81 7
3
= 7•7•7 = 49•7 = 343 Ex. 3a 1
5
Ex. 3b 2
3
•6
2
Solution: Solution: Write 1
5
in expanded form Write 2
3
•6
2
in expanded and multiply: form and multiply: 1
5
= 1•1•1•1•1 = 1•1•1•1 2
3
•6
2
= (2•2•2)•(6•6) = 1•1•1 = 1•1 = 1 = 8•36 = 288 Ex. 4a 10
1
Ex. 4b 10
2
Solution: Solution: 10
1
= 10 10
2
= 10•10 = 100

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31 Ex. 4c 10
3
Solution: 10
3
= 10•10•10 = 100•10 = 1000 Notice with powers of ten, we get a one followed by the numbers of zeros equal to the exponent. Ex. 4d 10
6
Ex. 4e 10
100
Solution: Solution: This will be equal to This will be equal to 1 followed by 6 zeros: 1 followed by 100 zeros: 1,000,000 10................0 |-100 zeros-| The number 10
100
is called a "googol." The mathematician that was first playing around with this number asked his nine-year old nephew to give it a name. There are even larger numbers like 10
googol
. This is a 1 followed by a googol number of zeros. If you were able to write 1 followed a googol number of zeros on a piece of paper, that paper could not be stuffed in the known universe! Objective b: Understanding and applying square roots. The square of a whole number is called a perfect square. So, 1, 4, 9, 16, 25 are perfect squares since 1 = 1
2
, 4 = 2
2
, 9 = 3
2
, 16 = 4
2
, and 25 = 5
2
. We use the idea of perfect squares to simplify square roots. The square root of a number a asks what number times itself is equal to a. For example, the square root of 25 is 5 since 5 times 5 is 25. The square root of a number a, denoted a , is a number whose square is a. So, 25 = 5. Simplify the following: Ex. 5a
49 Ex. 5b 144 Ex. 5c 0 Ex. 5d 625 Solution: a) 49 = 7 since 7
2
= 49. b) 144 = 12 since 12
2
= 144.

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32 c) 0 = 0 since 0
2
= 0. d) 625 = 25 since 25
2
= 625. Objective c: Understanding and applying the Order of Operations If you ever have done some cooking, you know how important it is to follow the directions to a recipe. An angel food cake will not come out right if you just mix all the ingredients and bake it in a pan. Without separating the egg whites from the egg yolks and whipping the eggs whites and so forth, you will end up with a mess. The same is true in mathematics; if you just mix the operations up without following the order of operations, you will have a mess. Unlike directions for making a cake that differ from recipe to recipe, the order of operations always stays the same. The order of operations are:
Order of Operations
1) Parentheses - Do operations inside of Parentheses ( ), [ ], { }, | | 2) Exponents including square roots. 3) Multiplication or Division as they appear from left to right. 4) Addition or Subtraction as they appear from left to right. A common phrase people like to use is:
Please
(Be careful with the My Dear and the
Excuse
Aunt Sally part. Multiplication does not
My Dear
precede Division and Division does not
Aunt Sally
precede multiplication; they are done as they appear from left to right. The same is true for addition and subtraction.)
Simplify the following:
Ex. 6 99 – 12 + 3 – 14 + 5 Solution: We need to add and subtract as they appear from left to right: 99 – 12 + 3 – 14 + 5 (#4-subtraction) = 87 + 3 – 14 + 5 (#4-addition) = 90 – 14 + 5 (#4-subtraction) = 76 + 5 (#4-addition) = 81

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33 Ex. 7 72 �� 9(4) �� 4 (5) Solution: 72 �� 9(4) �� 4 (5) (#2-exponents) We need to multiply and divide as they appear from left to right: = 72 �� 9(4) �� 2(5) (#3-division) = 8(4) �� 2(5) (#3-multiply) = 32 �� 2(5) (#3-division) = 16(5) (#3-multiply) = 80 Ex. 8 8•3 + 4•2 Solution: Since there are no parentheses or exponents, we start with step #3, multiply or divide as they appear from left to right: 8•3 + 4•2 (#3-multiplication) = 24 + 8 (#4-addition) = 32 Ex. 9 18 �� 3
2
• 16 + 4•5
2
Solution: Since there are no parentheses, we start with step #2, exponents: 18 �� 3
2
• 16 + 4•5
2
(#2-exponents) = 18 �� 9•4 + 4•25 (#3-division) = 2•4 + 4•25 (#3-multiplication) = 8 + 4•25 (#3-multiplication) = 8 + 100 (#4-addition) = 108 Ex. 10 ( 81 – 8)
3
+ 3•2
4
+ 0•5
2
Solution: ( 81 – 8)
3
+ 3•2
4
+ 0•5
2
(#1-parentheses, #2-exponents) = (9 – 8)
3
+ 3•2
4
+ 0•5
2
(#1-parentheses, #4-subtraction) = (1)
3
+ 3•2
4
+ 0•5
2
(#2-exponents) = 1 + 3•16 + 0•25 (#3-multiplication) = 1 + 48 + 0•25 (#3-multiplication) = 1 + 48 + 0 (#4-addition) = 49

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34 Ex. 11 9
2
– (24 – 12 �� 3) + 3•(5 – 2)
2
Solution: 9
2
– (24 – 12 �� 3) + 3•(5 – 2)
2
(#1-parentheses, #3-division) = 9
2
– (24 – 4) + 3•(5 – 2)
2
(#1-parentheses, #4-subtraction) = 9
2
– 20 + 3•(3)
2
(#2-exponents) = 81 – 20 + 3•9 (#3-multiplication) = 81 – 20 + 27 (#4-subtraction) = 61 + 27 (#4-addition) = 88 Ex. 12 9
1
+ 6(7 – 2) �� 3 – {8 – [3
2
– 16 ]} Solution: When there is a grouping symbol inside of another grouping symbol, work out the innermost set. So, we will work out [3
2
– 16 ] first: 9
1
+ 6(7 – 2) �� 3 – {8 – [3
2
– 16 ]} (#1-parent., #1-parent., #2-exp.) = 9
1
+ 6(7 – 2) �� 3 – {8 – [9 – 4]} (#1-parent., #1-parent., #4-subt.) = 9
1
+ 6(7 – 2) �� 3 – {8 – [5]} Notice that we can drop the [ ] since they are not needed. = 9
1
+ 6(7 – 2) �� 3 – {8 – 5} (#1-parent., #4-subt.) = 9
1
+ 6(5) �� 3 – {3} We can drop {} but not the (). = 9
1
+ 6(5) �� 3 – 3 (#2-exponents) = 9 + 6(5) �� 3 – 3 (#3-multiplication) = 9 + 30 �� 3 – 3 (#3-division) = 9 + 10 – 3 (#4-addition) = 19 – 3 (#4-subtraction) = 16 Objective d: Computing the mean (average). To find the average or mean of a set a numbers, we first add the numbers and then divided by the number of numbers. Ex. 13 Find the average of 79, 83, 91, 78, and 64. Solution: To find the average of a set of numbers, we add the numbers and then divide by the number of numbers: (79 + 83 + 91 + 78 + 64) �� 5 = (395) �� 5 = 79 So, the average is 79.

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