1
00:00:01,230 --> 00:00:05,600
Welcome to the presentation
on level one exponent rules.

2
00:00:05,600 --> 00:00:08,150
Let's get started
with some problems.

3
00:00:08,150 --> 00:00:12,870
So if I were to ask you what 2
-- that's a little fatter than

4
00:00:12,870 --> 00:00:15,080
I wanted it to be, but let's
keep it fat so it doesn't look

5
00:00:15,080 --> 00:00:20,260
strange -- 2 the third times --
and dot is another way of

6
00:00:20,260 --> 00:00:23,230
saying times -- if I were to
ask you what 2 to the third

7
00:00:23,230 --> 00:00:27,820
times 2 to the fifth is, how
would you figure that out?

8
00:00:27,820 --> 00:00:30,610
Actually, let me use a skinnier
pen because that does look bad.

9
00:00:30,610 --> 00:00:35,120
So, 2 to the third
times 2 to the fifth.

10
00:00:35,120 --> 00:00:37,610
Well there's one way that I
think you do know how to do it.

11
00:00:37,610 --> 00:00:42,150
You could figure out that
2 to the third is 9, and

12
00:00:42,150 --> 00:00:45,380
that 2 to the fifth is 32.

13
00:00:45,380 --> 00:00:46,840
And then you could
multiply them.

14
00:00:46,840 --> 00:00:54,010
And 8 times 32 is 240,
plus it's 256, right?

15
00:00:54,010 --> 00:00:55,530
You could do it that way.

16
00:00:55,530 --> 00:00:58,550
That's reasonable because it's
not that hard to figure out 2

17
00:00:58,550 --> 00:01:00,520
to the third is and what
2 to the fifth is.

18
00:01:00,520 --> 00:01:03,150
But if those were much larger
numbers this method might

19
00:01:03,150 --> 00:01:04,770
become a little difficult.

20
00:01:04,770 --> 00:01:08,520
So I'm going to show you using
exponent rules you can actually

21
00:01:08,520 --> 00:01:12,340
multiply exponentials or
exponent numbers without

22
00:01:12,340 --> 00:01:15,715
actually having to do as much
arithmetic or actually you

23
00:01:15,715 --> 00:01:18,120
could handle numbers much
larger than your normal math

24
00:01:18,120 --> 00:01:20,780
skills might allow you to.

25
00:01:20,780 --> 00:01:23,060
So let's just think what
2 to the third times

26
00:01:23,060 --> 00:01:24,670
2 to the fifth means.

27
00:01:24,670 --> 00:01:32,940
2 to the third is 2
times 2 times 2, right?

28
00:01:32,940 --> 00:01:35,200
And we're multiplying that
times 2 to the fifth.

29
00:01:35,200 --> 00:01:43,160
And that's 2 times 2
times 2 times 2 times 2.

30
00:01:43,160 --> 00:01:44,200
So what do we have here?

31
00:01:44,200 --> 00:01:47,870
We have 2 times 2 times
2, times 2 times 2 times

32
00:01:47,870 --> 00:01:49,780
2 times 2 times 2.

33
00:01:49,780 --> 00:01:52,640
Really all we're doing is we're
multiplying 2 how many times?

34
00:01:52,640 --> 00:01:58,920
Well, one, two, three, four,
five, six, seven, eight.

35
00:01:58,920 --> 00:02:03,410
So that's the same thing
as 2 to the eighth.

36
00:02:03,410 --> 00:02:05,050
Interesting.

37
00:02:05,050 --> 00:02:08,200
3 plus 5 is equal to 8.

38
00:02:08,200 --> 00:02:12,360
And that makes sense because 2
to the 3 is 2 multiplying by

39
00:02:12,360 --> 00:02:15,400
itself three times, to the
fifth is 2 multiplying by

40
00:02:15,400 --> 00:02:17,540
itself five times, and then
we're multiplying the two, so

41
00:02:17,540 --> 00:02:19,980
we're going to multiply
2 eight times.

42
00:02:19,980 --> 00:02:22,720
I hope I achieved my goal
of confusing you just now.

43
00:02:22,720 --> 00:02:23,580
Let's do another one.

44
00:02:26,130 --> 00:02:33,780
If I said 7 squared
times 7 to the fourth.

45
00:02:33,780 --> 00:02:36,550
That's a 4.

46
00:02:36,550 --> 00:02:42,180
Well, that equals 7 times 7,
right, that's 7 squared,

47
00:02:42,180 --> 00:02:44,430
times and now let's
do 7 to the fourth.

48
00:02:44,430 --> 00:02:50,090
7 times 7 times 7 times 7.

49
00:02:50,090 --> 00:02:53,780
Well now we're multiplying
7 by itself six times, so

50
00:02:53,780 --> 00:02:56,590
that equal 7 to the sixth.

51
00:02:56,590 --> 00:03:00,130
So in general, whenever I'm
multiplying exponents of the

52
00:03:00,130 --> 00:03:04,620
same base, that's key, I can
just add the exponents.

53
00:03:04,620 --> 00:03:12,520
So 7 to the hundredth power
times 7 to the fiftieth

54
00:03:12,520 --> 00:03:15,440
power, and notice this
is an example now.

55
00:03:15,440 --> 00:03:17,750
It would be very hard without
a computer to figure out what

56
00:03:17,750 --> 00:03:19,320
7 to the hundredth power is.

57
00:03:19,320 --> 00:03:22,190
And likewise, very hard without
a computer to figure out what

58
00:03:22,190 --> 00:03:24,050
7 to the fiftieth power is.

59
00:03:24,050 --> 00:03:32,730
But we could say that this is
equal to 7 to the 100 plus 50,

60
00:03:32,730 --> 00:03:37,790
which is equal to 7 to the 150.

61
00:03:37,790 --> 00:03:40,430
Now I just want to give you a
little bit of warning, make

62
00:03:40,430 --> 00:03:41,630
sure that you're multiplying.

63
00:03:41,630 --> 00:03:49,150
Because if I had 7 to the 100
plus 7 to the 50, there's

64
00:03:49,150 --> 00:03:50,590
actually very little
I could do here.

65
00:03:50,590 --> 00:03:54,440
I couldn't simplify
this number.

66
00:03:54,440 --> 00:03:56,710
But I'll throw out one to you.

67
00:03:56,710 --> 00:04:04,810
If I had 2 to the 8 times
2 to the 20, we know we

68
00:04:04,810 --> 00:04:06,570
can add these exponents.

69
00:04:06,570 --> 00:04:12,500
So that gives you 2
to the 28, right?

70
00:04:12,500 --> 00:04:20,820
What if I had 2 to the
8 plus 2 to the 8?

71
00:04:20,820 --> 00:04:22,890
This is a bit of a
trick question.

72
00:04:22,890 --> 00:04:25,030
Well I just said if
we're adding, we can't

73
00:04:25,030 --> 00:04:26,900
really do anything.

74
00:04:26,900 --> 00:04:28,530
We can't really simplify it.

75
00:04:28,530 --> 00:04:30,670
But there's a little trick
here that we actually have

76
00:04:30,670 --> 00:04:32,980
two 2 to the 8, right?

77
00:04:32,980 --> 00:04:35,080
There's 2 to the 8 times
1, 2 to the 8 times 2.

78
00:04:35,080 --> 00:04:41,240
So this is the same thing as 2
times 2 to the 8, isn't it?

79
00:04:41,240 --> 00:04:42,150
2 times 2 to the 8.

80
00:04:42,150 --> 00:04:44,940
That's just 2 to
the 8 plus itself.

81
00:04:44,940 --> 00:04:49,030
And 2 times to the 8, well
that's the same thing as 2 to

82
00:04:49,030 --> 00:04:53,170
the first times 2 to the 8.

83
00:04:53,170 --> 00:04:55,500
And 2 to the first times 2 to
the 8 by the same rule we just

84
00:04:55,500 --> 00:04:59,040
did is equal to 2 to the 9.

85
00:04:59,040 --> 00:05:01,080
So I thought I would just
throw that out to you.

86
00:05:01,080 --> 00:05:03,280
And it works even with
negative exponents.

87
00:05:03,280 --> 00:05:13,840
If I were to say 5 to the
negative 100 times 3 to the,

88
00:05:13,840 --> 00:05:18,370
say, 100 -- oh sorry, times
5 -- this has to be a 5.

89
00:05:18,370 --> 00:05:20,140
I don't know what my
brain was doing.

90
00:05:20,140 --> 00:05:25,150
5 to the negative 100 times
5 to the 102, that would

91
00:05:25,150 --> 00:05:27,890
equal 5 squared, right?

92
00:05:27,890 --> 00:05:30,930
I just take minus 100 plus 102.

93
00:05:30,930 --> 00:05:31,940
This is a 5.

94
00:05:31,940 --> 00:05:35,080
I'm sorry for that
brain malfunction.

95
00:05:35,080 --> 00:05:37,860
And of course, that equals 25.

96
00:05:37,860 --> 00:05:39,210
So that's the first
exponent rule.

97
00:05:39,210 --> 00:05:40,760
Now I'm going to show you
another one, and it kind of

98
00:05:40,760 --> 00:05:43,900
leads from the same thing.

99
00:05:43,900 --> 00:05:55,280
If I were to ask you what 2 to
the 9 over 2 to the 10 equals,

100
00:05:55,280 --> 00:05:56,940
that looks like that could
be a little confusing.

101
00:05:56,940 --> 00:06:00,720
But it actually turns out to be
the same rule, because what's

102
00:06:00,720 --> 00:06:03,110
another way of writing this?

103
00:06:03,110 --> 00:06:08,360
Well, we know that this is also
the same thing as 2 to the 9

104
00:06:08,360 --> 00:06:12,710
times 1 over 2 to
the 10, right?

105
00:06:12,710 --> 00:06:14,460
And we know 1 over 2 to the 10.

106
00:06:14,460 --> 00:06:18,700
Well, you could re-write right
this as 2 the 9 times 2 to

107
00:06:18,700 --> 00:06:20,850
the negative 10, right?

108
00:06:20,850 --> 00:06:25,270
All I did is I took 1 over 2 to
the 10 and I flipped it and I

109
00:06:25,270 --> 00:06:26,990
made the exponent negative.

110
00:06:26,990 --> 00:06:28,330
And I think you know
that already from

111
00:06:28,330 --> 00:06:30,660
level two exponents.

112
00:06:30,660 --> 00:06:33,090
And now, once again, we can
just add the exponents.

113
00:06:33,090 --> 00:06:39,300
9 plus negative 10 equals 2 to
the negative 1, or we could

114
00:06:39,300 --> 00:06:42,000
say that equals 1/2, right?

115
00:06:42,000 --> 00:06:44,730
So it's an interesting
thing here.

116
00:06:44,730 --> 00:06:48,110
Whatever is the bottom
exponent, you could put it in

117
00:06:48,110 --> 00:06:50,800
the numerator like we did here,
but turn it into a negative.

118
00:06:50,800 --> 00:06:53,760
So that leads us to the second
exponent rule, simplification

119
00:06:53,760 --> 00:06:59,860
is we could just say that this
equals 2 to the 9 minus 10,

120
00:06:59,860 --> 00:07:02,190
which equals 2 to
the negative 1.

121
00:07:02,190 --> 00:07:05,160
Let's do another
problem like that.

122
00:07:05,160 --> 00:07:16,400
If I said 10 to the 200 over
10 to the 50, well that

123
00:07:16,400 --> 00:07:23,640
just equals 10 to the 200
minus 50, which is 150.

124
00:07:23,640 --> 00:07:30,870
Likewise, if I had 7 to the
fortieth power over 7 to

125
00:07:30,870 --> 00:07:35,940
the negative fifth power,
this will equal 7 to the

126
00:07:35,940 --> 00:07:41,420
fortieth minus negative 5.

127
00:07:41,420 --> 00:07:46,230
So it equals 7 to
the forty-fifth.

128
00:07:46,230 --> 00:07:48,310
Now I want you to think about
that, does that make sense?

129
00:07:48,310 --> 00:07:54,480
Well, we could have re-written
this equation as 7 to the

130
00:07:54,480 --> 00:07:59,180
fortieth times 7 to
the fifth, right?

131
00:07:59,180 --> 00:08:02,806
We could have taken this 1 over
7 to the negative 5 and turn it

132
00:08:02,806 --> 00:08:06,640
into 7 to the fifth, and that
would also just be 7

133
00:08:06,640 --> 00:08:08,160
to the forty-five.

134
00:08:08,160 --> 00:08:10,810
So the second exponent rule I
just taught you actually is no

135
00:08:10,810 --> 00:08:12,390
different than that first one.

136
00:08:12,390 --> 00:08:14,810
If the exponent is in the
denominator, and of course, it

137
00:08:14,810 --> 00:08:18,210
has to be the same base and
you're dividing, you subtract

138
00:08:18,210 --> 00:08:20,570
it from the exponent
in the numerator.

139
00:08:20,570 --> 00:08:23,390
If they're both in the
numerator, as in this case, 7

140
00:08:23,390 --> 00:08:26,580
to the fortieth times 7 to the
fifth -- actually there's no

141
00:08:26,580 --> 00:08:29,370
numerator, but they're
essentially multiplying by each

142
00:08:29,370 --> 00:08:32,420
other, and of course, you have
to have the same base.

143
00:08:32,420 --> 00:08:35,690
Then you add the exponents.

144
00:08:35,690 --> 00:08:37,700
I'm going to add one variation
of this, and actually this is

145
00:08:37,700 --> 00:08:40,360
the same thing but it's a
little bit of a trick question.

146
00:08:40,360 --> 00:08:56,470
What is 2 to the 9
times 4 to the 100?

147
00:08:56,470 --> 00:08:58,190
Actually, maybe I shouldn't
teach this to you, you have

148
00:08:58,190 --> 00:08:59,480
to wait until I teach
you the next rule.

149
00:08:59,480 --> 00:09:01,900
But I'll give you
a little hint.

150
00:09:01,900 --> 00:09:09,570
This is the same thing as 2 the
9 times 2 squared to the 100.

151
00:09:09,570 --> 00:09:12,320
And the rule I'm going to teach
you now is that when you have

152
00:09:12,320 --> 00:09:15,610
something to an exponent and
then that number raised to

153
00:09:15,610 --> 00:09:18,930
an exponent, you actually
multiply these two exponents.

154
00:09:18,930 --> 00:09:24,980
So this would be 2 the
9 times 2 to the 200.

155
00:09:24,980 --> 00:09:27,020
And by that first rule
we learned, this would

156
00:09:27,020 --> 00:09:29,760
be 2 to the 209.

157
00:09:29,760 --> 00:09:31,140
Now in the next module
I'm going to cover

158
00:09:31,140 --> 00:09:31,940
this in more detail.

159
00:09:31,940 --> 00:09:34,650
I think I might have
just confused you.

160
00:09:34,650 --> 00:09:37,050
But watch the next video and
then after the next video I

161
00:09:37,050 --> 00:09:40,400
think you're going to be ready
to do level one exponent rules.

162
00:09:40,400 --> 00:09:41,930
Have fun.