1
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Multiply, expressing the product in scientific notation.

2
00:00:04,512 --> 00:00:05,883
So let's multiply first,

3
00:00:05,883 --> 00:00:09,491
and then let's get what we have into scientific notation.

4
00:00:09,491 --> 00:00:10,474
Actually, before we do that,

5
00:00:10,474 --> 00:00:13,380
let's remember what it means to be in scientific notation.

6
00:00:13,380 --> 00:00:14,941
So, to be in scientific notation,

7
00:00:14,941 --> 00:00:18,510
and actually, each of these numbers right here are in scientific notation,

8
00:00:18,510 --> 00:00:19,959
it's going to be the form

9
00:00:19,959 --> 00:00:23,015
"a" times ten to some power,

10
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where "a" can be greater than or equal to one,

11
00:00:27,829 --> 00:00:30,778
and it is going to be less than ten.

12
00:00:30,778 --> 00:00:33,551
So both of these numbers are greater than or equal to one,

13
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and less than ten,

14
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and they are being multiplied by some power of ten.

15
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Let's see how we can multiply this.

16
00:00:38,751 --> 00:00:42,443
So this over her is just the exact same thing as

17
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(I do this part in magenta)

18
00:00:43,931 --> 00:00:47,967
is the same as 9.1 times 10 to the sixth

19
00:00:47,967 --> 00:00:58,058
times ... (let me write it all out with a dot notation...

20
00:00:58,058 --> 00:00:59,393
make it a little more straight forward)

21
00:00:59,393 --> 00:01:09,421
So this is equal to 9.1 times 10 to the sixth

22
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times (in green) 3.2 times ten to the negative fifth power.

23
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Now, in multiplication, this comes from the associative property,

24
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which allows us to essentially remove these parentheses.

25
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It says, hey, you can multiply like that first,

26
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or you can actually multiply these guys first,

27
00:01:30,861 --> 00:01:32,297
you can re-associate them.

28
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And the commutative property tells us that

29
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we can rearrange this thing here.

30
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And what I want to rearrange is,

31
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I want to multiply the 9.1 times the 3.2 first,

32
00:01:40,632 --> 00:01:43,897
and then multiply that by 10 to the sixth times 10 to the negative five.

33
00:01:43,897 --> 00:01:46,453
So I'm just going to rearrange this using the commutative property.

34
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So this is the same thing as 9.1 times 3.2,

35
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and then I'm going to re-associate,

36
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so I'm going to do these first.

37
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And then that times

38
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10 to the 6th times 10 to the negative 5.

39
00:02:08,578 --> 00:02:11,857
And the reason this is useful, is that this is really easy to multiply.

40
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We have the same base here, base 10,

41
00:02:14,905 --> 00:02:16,804
so we can, and we're taking the product,

42
00:02:16,804 --> 00:02:18,534
so we can add the exponents.

43
00:02:18,534 --> 00:02:22,280
So this part right over here,

44
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10 to the 6th times 10 to the negative 5,

45
00:02:24,555 --> 00:02:27,611
that's going to be 10 to the 6 minus 5 power,

46
00:02:27,611 --> 00:02:30,413
so we're essentially just 10 to the first power,

47
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which is really just equal to ten.

48
00:02:32,266 --> 00:02:36,674
and that is going to be multiplied by 9.1 times 3.2

49
00:02:36,674 --> 00:02:37,935
Let me do that over here.

50
00:02:37,935 --> 00:02:43,457
9.1 times 3.2. So first, I'm going to ignore the decimal.

51
00:02:43,457 --> 00:02:45,667
I'm just going to treat it like 91 times 32.

52
00:02:45,667 --> 00:02:48,103
So I have 2 times 1 is 2.

53
00:02:48,103 --> 00:02:50,637
2 times 9 is 18.

54
00:02:50,637 --> 00:02:52,548
Take a zero her because I'm in the tenths place now,

55
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I'm really multiplying by 30,

56
00:02:55,221 --> 00:02:56,860
that's why my zero is there.

57
00:02:56,860 --> 00:03:00,035
I multiply 3 by 1 to get 3.

58
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And then 3 times 9 is 27.

59
00:03:03,412 --> 00:03:10,003
And so it is 2, (I'm adding here), 8 plus 3 is 11,

60
00:03:10,003 --> 00:03:17,938
carry/regroup the one, 1 plus 1 is two, 2 plus 7 is 9,

61
00:03:17,938 --> 00:03:19,651
and then I have a two here.

62
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91 times 32 is 2912,

63
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but I didn't multiply 91 by 32 here,

64
00:03:25,080 --> 00:03:28,249
I multiplied 9.1 times 3.2.

65
00:03:28,249 --> 00:03:32,071
So I want to count the number of digits behind the decimal point.

66
00:03:32,071 --> 00:03:35,332
And I have one, two digits behind the decimal point.

67
00:03:35,332 --> 00:03:38,634
And so I'll have to have two digits behind the decimal point in the answer.

68
00:03:38,634 --> 00:03:42,360
I'll stick the decimal right over there.

69
00:03:42,360 --> 00:03:45,067
So this part right over here,

70
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comes out to be 29.12.

71
00:03:48,159 --> 00:03:51,020
You might say, might feel like we're done.

72
00:03:51,020 --> 00:03:53,187
This kind of looks like scientific notation.

73
00:03:53,187 --> 00:03:56,051
I have a number times a power of ten.

74
00:03:56,051 --> 00:03:57,967
But remember,

75
00:03:57,967 --> 00:04:01,088
this number has to be greater than or equal to one,

76
00:04:01,088 --> 00:04:03,703
which it is, AND less than ten!

77
00:04:03,703 --> 00:04:05,390
But this number is not less than ten.

78
00:04:05,390 --> 00:04:07,764
Therefore, it is not scientific notation.

79
00:04:07,764 --> 00:04:09,548
So what we can do is, let's just write this number in

80
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scientific notation,

81
00:04:11,248 --> 00:04:13,595
and then we can use the multiply by ten part to multiply by

82
00:04:13,595 --> 00:04:14,987
this power of ten.

83
00:04:14,987 --> 00:04:23,903
So 29.12, this is the same thing as 2.912,

84
00:04:23,903 --> 00:04:26,104
notice, what did I do to get from there to there?

85
00:04:26,104 --> 00:04:28,533
I just moved the decimal to the left.

86
00:04:28,533 --> 00:04:30,470
Or, another way to think about it;

87
00:04:30,470 --> 00:04:33,021
if I wanted to go from here to there,

88
00:04:33,021 --> 00:04:34,835
what could I do to this?

89
00:04:34,835 --> 00:04:37,082
I would multiply by ten.

90
00:04:37,082 --> 00:04:38,569
I would move the decimal to the right;

91
00:04:38,569 --> 00:04:41,461
It would go from 2.912 to 29.12.

92
00:04:41,461 --> 00:04:43,287
So if I want to write this value,

93
00:04:43,287 --> 00:04:47,582
this is just this [2.912] times ten.

94
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29.12 is the same thing as 2.912 times 10.

95
00:04:52,469 --> 00:04:54,345
And now, this is in scientific notation,

96
00:04:54,345 --> 00:04:56,127
but that's just this part.

97
00:04:56,127 --> 00:04:58,027
We still have to multiply it by another 10.

98
00:04:58,027 --> 00:05:01,141
So, times another ten.

99
00:05:01,141 --> 00:05:02,928
And so, to finish up this problem,

100
00:05:02,928 --> 00:05:07,941
we get 2.912 times

101
00:05:07,941 --> 00:05:11,330
10 times 10, or 10 to the first times 10 to the first.

102
00:05:11,330 --> 00:05:12,472
Well, what's that?

103
00:05:12,472 --> 00:05:16,867
Well, that's just going to be 10 squared.

104
00:05:16,867 --> 00:05:20,537
So it is 2.912 times 10 to the second power.

105
00:05:20,537 --> 00:05:21,812
And we are done.

106
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Thanks You Sal! You are awesome! :)