1
00:00:01,980 --> 00:00:08,334
In this particular session,
we're going to be looking

2
00:00:08,334 --> 00:00:11,158
at indices or powers.

3
00:00:11,860 --> 00:00:15,292
Either name is used. Both mean

4
00:00:15,292 --> 00:00:22,168
the same. Basically there
a shorthand way of

5
00:00:22,168 --> 00:00:29,110
writing. Multiplications of the
same number. So here we have 4

6
00:00:29,110 --> 00:00:35,237
multiplied by itself three
times. So we write that as 4

7
00:00:35,237 --> 00:00:42,478
to the power three, so it's
three. That is the power or the

8
00:00:42,478 --> 00:00:49,089
index. That's the index or the
power. We can do this with

9
00:00:49,089 --> 00:00:55,795
letters, so we might have a
times a times a times a Times A

10
00:00:55,795 --> 00:01:01,064
and that's a multiplied by
itself five times. So we write

11
00:01:01,064 --> 00:01:04,417
that as A to the power 5.

12
00:01:05,090 --> 00:01:12,030
Do we have something like
2 X squared raised to

13
00:01:12,030 --> 00:01:15,500
the power four, let's say?

14
00:01:16,050 --> 00:01:21,341
Then that would mean two X
squared multiplied by two X

15
00:01:21,341 --> 00:01:26,151
squared multiplied by two X
squared multiplied by two X

16
00:01:26,151 --> 00:01:31,923
squared 1234 of them all
together. So we can do the tools

17
00:01:31,923 --> 00:01:39,138
together. 2 * 2 * 2 * 2.
That gives us 16 and X squared

18
00:01:39,138 --> 00:01:42,986
times by X squared is X to the

19
00:01:42,986 --> 00:01:48,660
power 4. Times by another X
squared is X to the power 6

20
00:01:48,660 --> 00:01:52,620
times by another. X squared is X
to the power 8.

21
00:01:54,280 --> 00:01:58,672
OK, we've got a notation. We've
got a way of writing something

22
00:01:58,672 --> 00:02:02,698
down. Now when mathematicians
have a notation and they got a

23
00:02:02,698 --> 00:02:07,456
way of writing something down,
they want to be able to use it

24
00:02:07,456 --> 00:02:11,482
for other purposes. So for
instance, what might A to the

25
00:02:11,482 --> 00:02:12,580
minus 2 mean?

26
00:02:13,080 --> 00:02:17,280
We know what A to the power two
would mean, but what about A to

27
00:02:17,280 --> 00:02:18,960
the minus two? What would that

28
00:02:18,960 --> 00:02:25,400
mean? What would something like
A to the power half me?

29
00:02:26,530 --> 00:02:33,000
What might something like A to
the power 0 mean?

30
00:02:33,620 --> 00:02:39,464
Well, we need some rules to
operate with an out of looking

31
00:02:39,464 --> 00:02:43,847
at these rules will find what
these particular notations

32
00:02:43,847 --> 00:02:49,482
actually mean. So let's begin
with our first rule. Supposing

33
00:02:49,482 --> 00:02:56,430
we have a cubed and we
want to multiply it by A

34
00:02:56,430 --> 00:02:58,746
squared, what's our result?

35
00:02:59,450 --> 00:03:05,988
Well, we know what I cubed is.
That means a times a times a

36
00:03:05,988 --> 00:03:12,526
three times times by A squared.
So that's a times by a on the

37
00:03:12,526 --> 00:03:18,597
end there, and altogether we've
got five of them A to the power

38
00:03:18,597 --> 00:03:24,870
5. And that suggests our very
first rule that if we're

39
00:03:24,870 --> 00:03:30,020
multiplying together expressions
such as these, then we add the

40
00:03:30,020 --> 00:03:37,230
indices and so if we have A
to the M times by A to

41
00:03:37,230 --> 00:03:44,440
the N and the result is A
to the N plus N, and that's

42
00:03:44,440 --> 00:03:45,985
our first rule.

43
00:03:46,690 --> 00:03:51,541
Let's have a look at our second
rule. Already done something

44
00:03:51,541 --> 00:03:56,833
like this previously. Supposing
we had A to the power four, and

45
00:03:56,833 --> 00:04:03,007
we want to raise it all to the
power three, and we know what

46
00:04:03,007 --> 00:04:09,181
that means. It means A to the
power four times by A to the

47
00:04:09,181 --> 00:04:12,268
power four times by A to the

48
00:04:12,268 --> 00:04:19,135
power 4. Now the first rule
tells us that we should add the

49
00:04:19,135 --> 00:04:26,065
indices together, so that's A to
the power twelve. 4 + 4 + 4.

50
00:04:26,990 --> 00:04:32,800
But twelve is 3 times by 4, so
that suggests to us that we

51
00:04:32,800 --> 00:04:36,950
should, perhaps, if we've got A
to the power M.

52
00:04:37,850 --> 00:04:44,670
Raised to the power N,
then the result we get

53
00:04:44,670 --> 00:04:49,444
by multiplying those two
together and that.

54
00:04:50,120 --> 00:04:52,808
Is our second rule.

55
00:04:53,710 --> 00:04:57,660
Let's now have a look.

56
00:04:58,340 --> 00:05:00,888
And our third rule.

57
00:05:01,630 --> 00:05:05,459
For this, let's take A to the

58
00:05:05,459 --> 00:05:12,698
7th. And let's divide it by
A to the power three or a

59
00:05:12,698 --> 00:05:18,297
cubed well, A to the 7th means
a multiplied by itself.

60
00:05:19,120 --> 00:05:26,180
Seven times.
Divided by so let's divide it

61
00:05:26,180 --> 00:05:31,980
by. A multiplied by itself 3
times and now we can begin to

62
00:05:31,980 --> 00:05:36,720
cancel some common factors. So
there's a common factor of A and

63
00:05:36,720 --> 00:05:40,670
again, there's another common
factor of A and again, there's

64
00:05:40,670 --> 00:05:45,410
another common factor of a. So
on the bottom here we've really

65
00:05:45,410 --> 00:05:53,020
got. 1 one and one suite. 1 *
1 is one and on the top a Times

66
00:05:53,020 --> 00:05:57,880
by a times by a Times by AA to
the power 4.

67
00:05:58,450 --> 00:06:04,794
But Seven takeaway three is A to
the power four, and so that

68
00:06:04,794 --> 00:06:12,114
gives us our third rule that if
we have A to the power M divided

69
00:06:12,114 --> 00:06:19,434
by A to the power N, we get
the result A to the power M

70
00:06:19,434 --> 00:06:25,846
minus N. And so there's
our third rule.

71
00:06:26,390 --> 00:06:31,837
OK, we got 3 rules. Let's see
what we can do with them.

72
00:06:34,340 --> 00:06:42,012
Let's have a look
at a cubed divided

73
00:06:42,012 --> 00:06:44,889
by a cubed.

74
00:06:45,800 --> 00:06:51,274
When we know the answer to that,
a cubed divided by a cube we're

75
00:06:51,274 --> 00:06:55,966
dividing something by itself. So
the answer is got to be 1.

76
00:06:57,130 --> 00:07:03,396
Fine, let's do it using our laws
of indices, our rules, and we

77
00:07:03,396 --> 00:07:10,626
can use Rule #3 for this that if
we want to do this, we subtract

78
00:07:10,626 --> 00:07:17,856
the indices. So that's A to the
power 3 - 3, which is A to

79
00:07:17,856 --> 00:07:19,302
the power 0.

80
00:07:20,240 --> 00:07:25,784
So what have we done? We've done
the same calculation in two

81
00:07:25,784 --> 00:07:30,404
different ways. We've done it
correctly in two different ways,

82
00:07:30,404 --> 00:07:36,410
so the answers that we get, even
if they look different, must be

83
00:07:36,410 --> 00:07:43,340
the same. And So what we have is
A to the power 0 equals 1.

84
00:07:43,370 --> 00:07:48,250
Our 4th result. If you like what
does this mean?

85
00:07:48,920 --> 00:07:53,252
Any fact it means that any
number raised to the power zero

86
00:07:53,252 --> 00:07:59,028
is one. So if we take two as we
raise it to the power zero, the

87
00:07:59,028 --> 00:08:00,111
answer is 1.

88
00:08:00,640 --> 00:08:07,264
If we take a million and
raise it to the power zero,

89
00:08:07,264 --> 00:08:09,472
the answer is 1.

90
00:08:10,940 --> 00:08:17,478
If we take something like a half
and raise it to the power zero,

91
00:08:17,478 --> 00:08:24,016
the answer is again one we take
minus six and raise it to the

92
00:08:24,016 --> 00:08:26,351
power zero. The answers one.

93
00:08:27,680 --> 00:08:30,328
If we take zero and raise it to

94
00:08:30,328 --> 00:08:35,773
the power 0. Well, it's a bit
complicated, so we'll leave that

95
00:08:35,773 --> 00:08:41,454
one on side for the moment. Just
bear in mind any number apart

96
00:08:41,454 --> 00:08:46,698
from zero when raised to the
power zero is equal to 1.

97
00:08:47,390 --> 00:08:54,438
Let's have a look
now at doing a

98
00:08:54,438 --> 00:08:55,319
division.

99
00:08:56,500 --> 00:09:01,373
Again. Let's take the example
that we use when we looked at

100
00:09:01,373 --> 00:09:05,009
law three, except let's turn it
round, let's do the division the

101
00:09:05,009 --> 00:09:10,050
other way about. A cubed divided
by A to the 7th.

102
00:09:10,720 --> 00:09:15,928
Well, you can set that out as we
did before, except it will be

103
00:09:15,928 --> 00:09:22,252
the other way up. So we have a
cubed is a Times by a times by a

104
00:09:22,252 --> 00:09:26,074
divided by. A multiplied

105
00:09:26,074 --> 00:09:29,680
by itself. 7.

106
00:09:30,390 --> 00:09:36,671
Times. Again, we can do
the canceling, canceling out the

107
00:09:36,671 --> 00:09:41,661
common factors, dividing top and
bottom by the common factors.

108
00:09:42,530 --> 00:09:49,656
So what do we have? One on
the top 1234 on the bottom A

109
00:09:49,656 --> 00:09:51,692
to the Power 4?

110
00:09:52,310 --> 00:09:59,942
We know we've done that right?
Let's use our third law, our

111
00:09:59,942 --> 00:10:05,666
third rule, and do it by
subtracting the indices.

112
00:10:06,130 --> 00:10:13,501
Well, three takeaway 7 is minus
four, so we've got A to the

113
00:10:13,501 --> 00:10:19,171
power minus four. So same
argument applies. We've done the

114
00:10:19,171 --> 00:10:24,486
calculation. Same calculation in
two different ways.

115
00:10:25,160 --> 00:10:30,250
We've done it correctly. We've
arrived at two different answers

116
00:10:30,250 --> 00:10:36,358
there for these two answers.
Have got to be the same. So

117
00:10:36,358 --> 00:10:42,975
one over 8 to the power four
is a till they minus 4.

118
00:10:43,940 --> 00:10:50,219
So a minus sign in the index
with the power means one over

119
00:10:50,219 --> 00:10:56,498
one over A to the power four.
Let's just develop that one a

120
00:10:56,498 --> 00:11:02,294
little bit. Let's just look at
one or two examples. So for

121
00:11:02,294 --> 00:11:07,607
instance, 2 to the power minus
two would be one over.

122
00:11:08,200 --> 00:11:14,977
Two Square, which of course
gives us a quarter.

123
00:11:15,880 --> 00:11:22,780
5 to the power minus one
is one over 5 to the

124
00:11:22,780 --> 00:11:26,805
one which is just one
over 5.

125
00:11:27,980 --> 00:11:30,005
One

126
00:11:30,005 --> 00:11:38,503
over. Hey.
Is A to the minus one turning

127
00:11:38,503 --> 00:11:40,779
it round working backwards?

128
00:11:41,520 --> 00:11:48,516
One over 7 squared would be
one over 49, but what about

129
00:11:48,516 --> 00:11:55,512
one over 7 to the minus
2 - 2 remember means one

130
00:11:55,512 --> 00:12:02,508
over 7 squared's. This is one
over one over 7 squared, and

131
00:12:02,508 --> 00:12:08,921
here we're dividing by a
fraction and to divide by a

132
00:12:08,921 --> 00:12:11,836
fraction, we know that we.

133
00:12:11,860 --> 00:12:18,820
Invert and multiply and so 7
times by 7 is 49 times

134
00:12:18,820 --> 00:12:25,200
by the one leaves us with
49 or just 7 squared.

135
00:12:26,460 --> 00:12:31,188
So some examples there. This is
probably the one that you need

136
00:12:31,188 --> 00:12:35,916
to remember and need to work
with most. It's the basic case

137
00:12:35,916 --> 00:12:41,038
and if you can remember that one
then they nearly all follow from

138
00:12:41,038 --> 00:12:44,584
that. So that's that one. Let's
now go on.

139
00:12:45,140 --> 00:12:47,540
And have a look.

140
00:12:47,620 --> 00:12:50,020
A tower 6 result.

141
00:12:50,550 --> 00:12:57,022
What do we mean
by A to the

142
00:12:57,022 --> 00:13:02,902
power 1/2? What's that mean? So
far we've been working with

143
00:13:02,902 --> 00:13:04,534
integers an with negative

144
00:13:04,534 --> 00:13:09,484
numbers. What about A to the
power 1/2 well?

145
00:13:10,230 --> 00:13:16,630
Supposing we had A to the P and
we multiplied it by A to the P,

146
00:13:16,630 --> 00:13:19,430
the answer we got was just a.

147
00:13:20,030 --> 00:13:23,726
Just a That's A to the

148
00:13:23,726 --> 00:13:27,246
power one. And a Times

149
00:13:27,246 --> 00:13:33,642
by a. Each with a P on A to the
P times by A to the P using our

150
00:13:33,642 --> 00:13:35,644
rule would be A to the 2P.

151
00:13:36,830 --> 00:13:43,648
So 2P must be the same as one.
In other words, P is 1/2.

152
00:13:44,660 --> 00:13:49,456
What do we have some sort of
interpretation for? This two

153
00:13:49,456 --> 00:13:53,816
numbers, identical that multiply
together to give a? Well, that's

154
00:13:53,816 --> 00:13:55,996
the square root, isn't it?

155
00:13:56,660 --> 00:14:02,420
It's a square root. It's like 7
times by 7 equals 49.

156
00:14:03,390 --> 00:14:05,770
So if we take that one on.

157
00:14:06,900 --> 00:14:14,460
7 times by 7 is 49. What
we've got then is 49 to the

158
00:14:14,460 --> 00:14:22,020
half is equal to 7, so ETA
the half is equal to the square

159
00:14:22,020 --> 00:14:23,640
root of A.

160
00:14:24,190 --> 00:14:31,613
A to the third would be equal
to the cube root of A.

161
00:14:32,210 --> 00:14:38,755
So if we were asked what
is 16 to the quarter?

162
00:14:38,760 --> 00:14:42,990
What we're asking is what
number, when multiplied by

163
00:14:42,990 --> 00:14:48,630
itself four times, gives us 16.
A fairly obvious choice for that

164
00:14:48,630 --> 00:14:51,520
is 2. What

165
00:14:51,520 --> 00:14:57,612
about 81? To
the half? Well, that's the

166
00:14:57,612 --> 00:15:03,156
square root of 81, and the
square root of 81 is 9.

167
00:15:03,960 --> 00:15:10,872
What about 243
and will make

168
00:15:10,872 --> 00:15:14,328
that to the

169
00:15:14,328 --> 00:15:19,990
5th? What number, when
multiplied by itself five times,

170
00:15:19,990 --> 00:15:27,415
will give us 240 three? Well, as
we look at this, we can see it

171
00:15:27,415 --> 00:15:33,850
divides by three and three's
into that give us 81, so we know

172
00:15:33,850 --> 00:15:36,820
this is 3 times by 81.

173
00:15:37,410 --> 00:15:41,178
81 we know.

174
00:15:41,790 --> 00:15:45,550
Is 9 times by 9.

175
00:15:45,550 --> 00:15:52,198
And each of
those nines is

176
00:15:52,198 --> 00:15:55,522
3 times by

177
00:15:55,522 --> 00:16:02,400
three. Which means the number
that we want is just three.

178
00:16:03,260 --> 00:16:09,354
Noticing doing this, how
important it is to be able to

179
00:16:09,354 --> 00:16:16,002
recognize what numbers are made
up of to be able to recognize

180
00:16:16,002 --> 00:16:23,758
that 16 is 2 to the power,
four that it's also 4 to the

181
00:16:23,758 --> 00:16:31,514
power to the nine is 3 squared.
That 81 is 9 squared and also

182
00:16:31,514 --> 00:16:34,284
3 to the power 4.

183
00:16:34,320 --> 00:16:40,390
You'll find calculations much,
much easier if you can recognize

184
00:16:40,390 --> 00:16:47,067
in numbers their composition as
powers of simple numbers such as

185
00:16:47,067 --> 00:16:50,709
two and three, four, and five.

186
00:16:51,220 --> 00:16:53,767
Once you've got those firmly
fixed in your mind.

187
00:16:54,820 --> 00:16:58,325
This sort of calculation becomes

188
00:16:58,325 --> 00:17:01,146
relatively straightforward. 1

189
00:17:01,146 --> 00:17:07,513
final result. If
we now know what A to the half

190
00:17:07,513 --> 00:17:12,231
isn't A to the third and eight
to the quarter, what do we mean

191
00:17:12,231 --> 00:17:14,590
if we take A to the 3/4?

192
00:17:15,800 --> 00:17:21,300
Well, the quarters alright.
Let's split that up. That means

193
00:17:21,300 --> 00:17:24,050
A to the power 1/4.

194
00:17:24,970 --> 00:17:29,254
Cubed and what we're doing is
we're using this result that A

195
00:17:29,254 --> 00:17:34,966
to the M raised to the power N
is A to the MN. In other words,

196
00:17:34,966 --> 00:17:38,893
we're using our 2nd result to be
able to do that.

197
00:17:40,190 --> 00:17:46,994
So let's have a look at an
example using this. Supposing we

198
00:17:46,994 --> 00:17:53,231
take 60, we say we want 16
to the power 3/4.

199
00:17:53,240 --> 00:17:56,726
Then that 16 to the power

200
00:17:56,726 --> 00:18:03,978
quarter. To be cubed, so we
look at this first 16 to the

201
00:18:03,978 --> 00:18:09,577
Power 1/4 is 2. That's the
number when multiplied by itself

202
00:18:09,577 --> 00:18:15,176
four times will give us 16
raised to the power 3.

203
00:18:15,190 --> 00:18:19,600
And that of course is 8 because
that means 2 * 2 * 2.

204
00:18:20,220 --> 00:18:25,104
But we can think of this another
way, because Eminem can be

205
00:18:25,104 --> 00:18:29,988
interchanged. Currently we could
write this as A to the power N

206
00:18:29,988 --> 00:18:32,023
raised to the power M.

207
00:18:32,040 --> 00:18:39,359
That would be just the same
result, so we can think of this

208
00:18:39,359 --> 00:18:44,989
in a slightly different way.
Let's take a different example.

209
00:18:44,989 --> 00:18:46,678
Let's take 8.

210
00:18:46,690 --> 00:18:49,790
To the power 2/3.

211
00:18:50,290 --> 00:18:56,605
Now, the way that we had thought
about that was to do 8 to the

212
00:18:56,605 --> 00:18:58,908
power third. Then square it.

213
00:18:59,490 --> 00:19:05,665
And we know that 8 to the
power 1:30 is 2 and 2

214
00:19:05,665 --> 00:19:08,040
squared is there for four.

215
00:19:09,130 --> 00:19:15,472
We don't have to think of it
like that. We can think of it

216
00:19:15,472 --> 00:19:20,455
as 8 squared raised to the
Power, 1/3 eight squared we

217
00:19:20,455 --> 00:19:27,250
know is 64 and now we have to
take it to the power 1/3. We

218
00:19:27,250 --> 00:19:32,233
need the cube root. We need a
number that when multiplied

219
00:19:32,233 --> 00:19:37,669
by itself three times gives
us 64 and that number is 4.

220
00:19:38,750 --> 00:19:41,410
These two are equivalent.

221
00:19:41,930 --> 00:19:47,690
So the different interpretations
that we've got are the same.

222
00:19:47,690 --> 00:19:52,060
Writing it down algebraically,
what we're saying is that we

223
00:19:52,060 --> 00:19:58,941
have. A letter of variable to
the power P over Q. Then we can

224
00:19:58,941 --> 00:20:05,346
write that in one of two ways.
We can write it as either A to

225
00:20:05,346 --> 00:20:06,627
the power P.

226
00:20:07,140 --> 00:20:14,308
Find the cube root of it, which
might be written as A to the

227
00:20:14,308 --> 00:20:21,476
power P. Find the cube root, or
it might be written as a. Let's

228
00:20:21,476 --> 00:20:27,620
take the cube root and raise it
all to the power P.

229
00:20:28,160 --> 00:20:34,656
So we might have a find the cube
root of it and raise it all to

230
00:20:34,656 --> 00:20:35,874
the power P.

231
00:20:36,420 --> 00:20:39,460
Either result is exactly the

232
00:20:39,460 --> 00:20:46,286
same. Now, to conclude, let's
just have a look at some very

233
00:20:46,286 --> 00:20:48,230
basic simple calculations using

234
00:20:48,230 --> 00:20:54,536
indices. 2X to the minus 1/4 and
our objective here is to write

235
00:20:54,536 --> 00:20:59,732
it with a positive index. Well,
the two doesn't have an index

236
00:20:59,732 --> 00:21:04,928
attached to it at all, so
nothing happens to the two just

237
00:21:04,928 --> 00:21:10,990
stays it is X to the minus 1/4.
The minus sign means one over

238
00:21:10,990 --> 00:21:16,619
right it in the denominator, so
we write it down there. So we

239
00:21:16,619 --> 00:21:19,650
have two over X to the power

240
00:21:19,650 --> 00:21:25,170
quarter. 4X to the minus 2A
cubed. Well, nothing wrong with

241
00:21:25,170 --> 00:21:31,194
the four and nothing wrong with
the A cubed at perfectly normal

242
00:21:31,194 --> 00:21:34,206
so they stay as they are.

243
00:21:34,750 --> 00:21:36,330
X to the minus 2.

244
00:21:37,230 --> 00:21:44,706
A minus in the index and so
that means one over, so it's 4A

245
00:21:44,706 --> 00:21:46,842
cubed over X squared.

246
00:21:48,850 --> 00:21:54,546
One over 4A to the
power minus 2.

247
00:21:55,280 --> 00:21:58,838
Well, we met this one before.

248
00:21:59,340 --> 00:22:06,132
If you remember, we actually
looked at one over 7 to the

249
00:22:06,132 --> 00:22:13,435
minus 2. And that was
one over one over 7 squared

250
00:22:13,435 --> 00:22:20,455
and because we were dividing by
a fraction, we said we had

251
00:22:20,455 --> 00:22:27,475
to invert and multiply. And so
we ended up with 7 squared.

252
00:22:28,440 --> 00:22:33,060
This is no different. As a
letter there instead of a

253
00:22:33,060 --> 00:22:37,260
number, the result is going to
be exactly the same.

254
00:22:37,830 --> 00:22:44,942
So the four stays where it is
one over 4 and this becomes a

255
00:22:44,942 --> 00:22:50,530
squared and to write it in a
more simple, tidier fashion,

256
00:22:50,530 --> 00:22:53,070
it's A squared over 4.

257
00:22:54,130 --> 00:23:01,800
A to the minus 1/3
times by two A to

258
00:23:01,800 --> 00:23:04,868
the minus 1/2 equals.

259
00:23:05,480 --> 00:23:13,144
Two, we can just write down A to
the minus 1/3 times by 8 in the

260
00:23:13,144 --> 00:23:18,892
minus 1/2 hour. First job is to
add those indices together, so

261
00:23:18,892 --> 00:23:21,287
minus 1/3 plus minus 1/2.

262
00:23:21,930 --> 00:23:29,602
He's going to give us 2A to
the power now. A third and a

263
00:23:29,602 --> 00:23:36,726
half is 5 six. So with the
minus signs, that is minus five

264
00:23:36,726 --> 00:23:43,302
sixths, and so that's two over A
to the power five sixths.

265
00:23:44,270 --> 00:23:50,774
2A to the
minus 2 divided

266
00:23:50,774 --> 00:23:57,278
by A to
the minus three

267
00:23:57,278 --> 00:24:04,794
over 2. Well, that's
2A to the minus 2 divided

268
00:24:04,794 --> 00:24:08,352
by A to the minus three

269
00:24:08,352 --> 00:24:14,584
over 2. Let's remember what our
rules said that if we're

270
00:24:14,584 --> 00:24:18,958
dividing things like this, we
actually subtract the indices.

271
00:24:18,958 --> 00:24:25,762
So this is 2A to the minus 2
minus minus three over two. Of

272
00:24:25,762 --> 00:24:31,108
course, that means effectively
we've got to add on to three

273
00:24:31,108 --> 00:24:38,884
over 2, so we get 2A to the
minus 2 + 3 over 2 inches minus

274
00:24:38,884 --> 00:24:40,828
1/2, so that's 2A.

275
00:24:40,840 --> 00:24:44,240
Over A to the half.

276
00:24:44,790 --> 00:24:52,326
Now then we take
something that looks a

277
00:24:52,326 --> 00:24:55,152
little bit complicated.

278
00:24:55,820 --> 00:25:03,370
So what do we got
here? We've got the cube

279
00:25:03,370 --> 00:25:10,920
root of A squared times
by the square root of

280
00:25:10,920 --> 00:25:17,560
a cube. First of all, let's
write these as indices. This is

281
00:25:17,560 --> 00:25:23,397
the cube root of A squared. So
that means a squared and take

282
00:25:23,397 --> 00:25:27,887
the cube root. You know it's A
to the 2/3.

283
00:25:28,520 --> 00:25:35,540
Times by a now this is
a cubed. Take the square root

284
00:25:35,540 --> 00:25:39,050
so that's a cubed. Take the

285
00:25:39,050 --> 00:25:46,396
square root. And we can see
now what we need to do is add

286
00:25:46,396 --> 00:25:52,350
together these two indices. So
that's 2/3 + 3 over 2 and adding

287
00:25:52,350 --> 00:25:57,846
fractions together. That will
give us A to the power 13 over

288
00:25:57,846 --> 00:26:03,800
6 awkward number will just leave
it like that for now. Now let's

289
00:26:03,800 --> 00:26:09,296
have a look at some calculations
using numbers. This time 16 to

290
00:26:09,296 --> 00:26:13,876
the Power 3/4. Now again, we've
already done this one.

291
00:26:14,810 --> 00:26:18,560
That 16 to the power 1/4.

292
00:26:19,120 --> 00:26:23,719
Cubed what number, when
multiplied by itself four times,

293
00:26:23,719 --> 00:26:30,362
gives us 16? That's two raised
to the power. Three gives us 8

294
00:26:30,362 --> 00:26:31,895
for our answer.

295
00:26:32,680 --> 00:26:39,583
4 to the power minus five over
2. Let's deal with the minus

296
00:26:39,583 --> 00:26:45,424
sign first. Minus sign means one
over. Put it until the

297
00:26:45,424 --> 00:26:48,610
denominator, 4 to the power 5

298
00:26:48,610 --> 00:26:55,080
over 2. One over 4 to
the half to the power five

299
00:26:55,080 --> 00:27:00,924
notice each time I'm doing the
root bit first, the 4th root

300
00:27:00,924 --> 00:27:05,794
here, the square root here.
That's because by doing the

301
00:27:05,794 --> 00:27:11,151
route, I get the number smaller.
I can handle the arithmetic

302
00:27:11,151 --> 00:27:18,456
easier, so this is now one over
the square root of 4 is 2 raised

303
00:27:18,456 --> 00:27:20,404
to the power 5.

304
00:27:20,440 --> 00:27:24,388
And that's one over 32 again.
Notice I wrote it straight down

305
00:27:24,388 --> 00:27:28,336
without seeming to work it out.
That's because I know what it

306
00:27:28,336 --> 00:27:32,284
is. And again, it's another one
of these relationships. 2 to the

307
00:27:32,284 --> 00:27:36,890
power, three is 8 two to the
power, five is 32 to the power,

308
00:27:36,890 --> 00:27:40,838
six is 64 that really, if you
can learn and get become

309
00:27:40,838 --> 00:27:44,128
familiar with you, find this
kind of calculation much easier.

310
00:27:45,610 --> 00:27:53,420
Here's another one. 125 seems
a very big number to

311
00:27:53,420 --> 00:28:00,546
the 2/3. Equals 125
to the power of

312
00:28:00,546 --> 00:28:03,054
3rd all squared.

313
00:28:03,750 --> 00:28:09,067
Now, even if we've never met 125
before, in these terms, one of

314
00:28:09,067 --> 00:28:15,611
things we ought to be aware of
is it ends in a 5, so it must

315
00:28:15,611 --> 00:28:20,519
divide by 5. So it's a fair
guess that 5 multiplied by

316
00:28:20,519 --> 00:28:25,836
itself three times would give us
125, and indeed it does so to

317
00:28:25,836 --> 00:28:30,744
the third power, 125 to the
third power is 5. The number

318
00:28:30,744 --> 00:28:31,971
multiplied by itself.

319
00:28:32,640 --> 00:28:39,195
Three times that gives us 125 is
5 and now we just want to square

320
00:28:39,195 --> 00:28:41,817
that and that obviously gives us

321
00:28:41,817 --> 00:28:49,580
25. Take 8 - 2/3
again. Let's deal with the minus

322
00:28:49,580 --> 00:28:55,295
sign. That is one over 8
to the 2/3.

323
00:28:55,860 --> 00:29:01,349
Again, let's deal with the
third bit, the root bit first.

324
00:29:01,349 --> 00:29:05,840
What number, when multiplied
by itself three times, would

325
00:29:05,840 --> 00:29:12,826
give us eight that must be 2
Ann. We need to square it and

326
00:29:12,826 --> 00:29:14,822
so that becomes 1/4.

327
00:29:18,490 --> 00:29:24,774
We take
one over

328
00:29:24,774 --> 00:29:31,058
25 to
the minus

329
00:29:31,058 --> 00:29:38,707
2. Well, remember we
saw what this happened with the

330
00:29:38,707 --> 00:29:46,063
one over 7 to the minus
two this became 25 squared and

331
00:29:46,063 --> 00:29:48,515
that is simply 625.

332
00:29:50,190 --> 00:29:53,610
243 to

333
00:29:53,610 --> 00:30:01,550
the 3/5.
Let's do the 5th bit 1st

334
00:30:01,550 --> 00:30:07,900
and then the QB. Well 243,
that's three multiplied by

335
00:30:07,900 --> 00:30:15,520
itself five times. So that is
a three race to the power

336
00:30:15,520 --> 00:30:18,060
3, and that's 27.

337
00:30:19,390 --> 00:30:26,644
One last example,
let's take 81

338
00:30:26,644 --> 00:30:30,271
and to the

339
00:30:30,271 --> 00:30:37,174
minus 3/4.
OK, minus sign means

340
00:30:37,174 --> 00:30:43,992
one over. Where dividing
by a fraction to

341
00:30:43,992 --> 00:30:50,728
divide by a fraction,
we invert and multiply,

342
00:30:50,728 --> 00:30:57,464
so this becomes 16
over 81 raised to

343
00:30:57,464 --> 00:30:59,990
the power 3/4.

344
00:31:00,050 --> 00:31:07,050
Equals 16 over 81.
First of all to

345
00:31:07,050 --> 00:31:10,550
the quarter and then.

346
00:31:10,560 --> 00:31:15,824
Raised to the power three to the
quarter, this is 2 to the power

347
00:31:15,824 --> 00:31:21,088
four on top 16 is 2 to the power
four. So that's a 2.

348
00:31:21,860 --> 00:31:28,548
And 81 is 3 to the power four,
so that's a 3 and we need to

349
00:31:28,548 --> 00:31:32,310
cube that and that gives us
eight over 27th.

350
00:31:33,130 --> 00:31:37,243
So working with these indices
shouldn't really be too

351
00:31:37,243 --> 00:31:42,270
difficult. You need to remember
how numbers are made up. You

352
00:31:42,270 --> 00:31:48,211
need to have in your mind's eye
that 243 is 3 multiplied by

353
00:31:48,211 --> 00:31:54,152
itself. 5 * 243 is 3 to the
power five, and similarly with

354
00:31:54,152 --> 00:31:56,437
numbers like 30, two, 64128.

355
00:31:57,750 --> 00:32:02,958
If you can carry those around in
your head and you will do with

356
00:32:02,958 --> 00:32:07,422
practice, you get to know them
quite well. Then this kind of

357
00:32:07,422 --> 00:32:11,142
calculation should become very
easy to remember. Deal with the

358
00:32:11,142 --> 00:32:15,606
minus sign first. By and large.
Get that to be positive, then

359
00:32:15,606 --> 00:32:19,698
deal with the root sign that
gets the number. Work down.

360
00:32:19,698 --> 00:32:20,814
Keeps it small.

361
00:32:21,740 --> 00:32:24,435
They can be tricky, but we
hope we've shown that they

362
00:32:24,435 --> 00:32:25,170
can be mastered.