1
00:00:05,000 --> 00:00:07,080
The other option is let's kind of check this out.

2
00:00:07,160 --> 00:00:09,920
49 is the same thing as seven squared, right?

3
00:00:12,980 --> 00:00:20,085
Okay, and then using our laws of exponents, a base to a power raised to a power, we

4
00:00:20,085 --> 00:00:20,480
multiply

5
00:00:20,480 --> 00:00:23,140
them, and it's 7.

6
00:00:25,260 --> 00:00:28,900
Let's try another one for fun.

7
00:00:33,540 --> 00:00:46,220
I could write 25 as what? 5 squared, which is 5 to the 2 times one half, which is 5.

8
00:00:47,180 --> 00:00:50,460
Okay, which is the same thing as the square root of 25.

9
00:00:55,280 --> 00:00:58,040
Uh, let's try a different one.

10
00:01:10,600 --> 00:01:11,660
What number squared is 30?

11
00:01:14,320 --> 00:01:20,118
It's not a perfect square, but there is a number, when you square it is 30 and it

12
00:01:20,118 --> 00:01:20,868
is.

13
00:01:23,060 --> 00:01:29,887
It's the square root of 30. That's the definition of a number that when you square

14
00:01:29,887 --> 00:01:30,740
it equals

15
00:01:30,740 --> 00:01:33,440
that number. The definition is the square root.

16
00:01:36,560 --> 00:01:42,505
And that is 2 times 1 half, which is the square root of 30 to the power of 1, which

17
00:01:42,505 --> 00:01:43,100
is the

18
00:01:43,100 --> 00:01:44,000
square root of 30.

19
00:01:44,780 --> 00:01:51,680
Okay, so x to the 1 half power is exactly the same thing as the square root of x.

20
00:01:51,680 --> 00:01:52,760
That's the one-half power.

21
00:01:52,860 --> 00:01:54,200
What about a one-third power?

22
00:01:54,200 --> 00:01:57,240
Why is that?

23
00:02:01,780 --> 00:02:03,660
What number cubed is 8?

24
00:02:05,040 --> 00:02:05,940
Two.

25
00:02:08,700 --> 00:02:09,600
Oops.

26
00:02:11,000 --> 00:02:13,420
Which is two to the three times one third.

27
00:02:14,380 --> 00:02:16,120
Which is two to the first, which is two.

28
00:02:18,000 --> 00:02:18,380
Okay.

29
00:02:18,380 --> 00:02:26,380
which is the cube root of 8. The number that when you cube it is 8 is the cube root

30
00:02:26,380 --> 00:02:28,380
of 8, which is 2.

31
00:02:33,760 --> 00:02:40,240
What about a number that doesn't... let's do this... 10 to the one third.

32
00:02:40,240 --> 00:02:51,720
now. The number that when you cube it is 10 is by definition the cube root of 10.

33
00:02:53,780 --> 00:03:03,440
And this becomes the cube root of 10. So X to the 1 third is exactly the cube

34
00:03:03,440 --> 00:03:12,421
root of x. Is that going to work with every single denominator? One fourth? One

35
00:03:12,421 --> 00:03:13,171
fifth?

36
00:03:14,600 --> 00:03:19,300
One sixth? Yes. Okay, like,

37
00:03:30,160 --> 00:03:31,700
81 to the one-fourth.

38
00:03:39,080 --> 00:03:41,520
What to the power of four is 81?

39
00:03:43,000 --> 00:03:43,900
Three.

40
00:03:49,660 --> 00:03:53,760
which is the same thing as the fourth root of 81.

41
00:03:57,660 --> 00:04:05,300
X to the power of one over any number b is the bth root of that number.

42
00:04:05,300 --> 00:04:12,900
Uh oh, now it's not just one over a number in the next one.

43
00:04:12,900 --> 00:04:16,480
I have 64 to the two-thirds power.

44
00:04:22,800 --> 00:04:28,360
Is this possible yeah, how are we gonna do this or deal with this?

45
00:04:36,040 --> 00:04:38,720
Okay, so one of our laws

46
00:04:40,440 --> 00:04:45,260
2 thirds is the same thing as 1 third times 2.

47
00:04:46,180 --> 00:04:48,480
Well, that's not a law; that's just multiplication.

48
00:04:50,500 --> 00:04:59,560
One of our laws is x to the a to the power of b is equal to x to the a times b.

49
00:05:01,500 --> 00:05:04,780
So here I got this part right here.

50
00:05:05,700 --> 00:05:10,680
So I'm going to write this as 64 to the one third to the power of 2.

51
00:05:13,720 --> 00:05:18,480
We know the one third power is of 64 is 4.

52
00:05:23,060 --> 00:05:25,900
And we have 4 squared is 16.

53
00:05:31,020 --> 00:05:36,780
And we could have done it this way as well.

54
00:05:38,400 --> 00:05:48,560
64 to the 2 times 1 third, which is 64 squared to the 1 third.

55
00:05:51,020 --> 00:05:52,660
What is 64 squared?

56
00:05:53,300 --> 00:05:54,200
Anybody have a calculator?

57
00:05:57,080 --> 00:06:02,280
It said 4,096.

58
00:06:02,520 --> 00:06:03,200
Oh yeah.

59
00:06:03,200 --> 00:06:04,100
Somebody do it on the calculator.

60
00:06:07,740 --> 00:06:13,040
I think it's 4,096.

61
00:06:13,340 --> 00:06:14,240
I got it.

62
00:06:14,460 --> 00:06:15,420
4,096.

63
00:06:16,100 --> 00:06:23,000
And then what is the cube root of 4,096?

64
00:06:23,000 --> 00:06:30,860
Yeah, it better be exactly 16 where math is dead if that's not the case

65
00:06:33,760 --> 00:06:38,400
Okay notice it's a lot easier to first do the root

66
00:06:39,560 --> 00:06:47,400
Because it makes a small number and then we do the power a

67
00:06:47,400 --> 00:06:51,755
Lot easier than making the number a lot bigger and then who knows what the cube root

68
00:06:51,755 --> 00:06:52,780
of that number is

69
00:06:53,000 --> 00:06:54,120
OK, that gets difficult.

70
00:06:56,320 --> 00:07:05,220
So in general, x to the a over b can be written two different ways.

71
00:07:06,560 --> 00:07:12,640
It could be the b-th root of x, raised to the power of a.

72
00:07:17,760 --> 00:07:20,800
Or the b-th root of x to the power of a.

73
00:07:20,800 --> 00:07:21,820
Let's try this one.

74
00:07:30,760 --> 00:07:34,380
So that's going to be the what root?

75
00:07:38,060 --> 00:07:39,540
Fifth root.

76
00:07:39,820 --> 00:07:41,560
Whatever the denominator is, is the root.

77
00:07:44,280 --> 00:07:46,120
To the power of three.

78
00:07:48,880 --> 00:07:50,760
What is the fifth root of 32?

79
00:07:52,740 --> 00:07:53,640
Two.

80
00:07:53,880 --> 00:08:08,020
Two cubed is 8. The other way to do it is to take our base, which is

81
00:08:08,080 --> 00:08:11,300
32 and just start factoring it.

82
00:08:16,400 --> 00:08:19,880
2 to the 4, 2 to the 2.

83
00:08:21,480 --> 00:08:23,000
So that's 2 to the 5th.

84
00:08:27,340 --> 00:08:27,940
Just

85
00:08:27,940 --> 00:08:33,832
keep it as exponents, don't mess with anything. A base with an exponent raised to an

86
00:08:33,832 --> 00:08:34,200
exponent

87
00:08:34,200 --> 00:08:47,340
that's 2 to the 5 times 3 fifths and what is 5 times 3 fifths 2 cubed which is 8 so

88
00:08:47,340 --> 00:08:54,380
you leave it all as exponents once you factor the whatever the base is and then

89
00:08:54,380 --> 00:08:57,000
to just use the rules of exponents to multiply them.

90
00:08:57,000 --> 00:08:58,120
The square root of 49 is?

91
00:08:58,420 --> 00:08:59,320
Seven.

92
00:08:59,400 --> 00:09:00,300
Seven.

93
00:09:01,340 --> 00:09:02,360
Is it also negative 7?

94
00:09:03,820 --> 00:09:04,800
OK, this is very important.

95
00:09:05,060 --> 00:09:05,960
Let's talk about a few things.

96
00:09:10,480 --> 00:09:14,280
First up, is there a, what's the root here?

97
00:09:14,880 --> 00:09:15,780
2.

98
00:09:15,900 --> 00:09:21,720
Or the, OK, that's called the index, or the indices index.

99
00:09:22,500 --> 00:09:23,400
Tell you the root.

100
00:09:23,700 --> 00:09:25,940
Square root obviously has an invisible 2 there.

101
00:09:27,200 --> 00:09:30,340
This is called the radical symbol.

102
00:09:34,240 --> 00:09:35,960
Radical sign. Okay.

103
00:09:36,180 --> 00:09:37,900
And the thing inside it.

104
00:09:38,940 --> 00:09:41,040
It's called the radicand.

105
00:09:45,620 --> 00:09:51,640
Okay. Whenever you see just the radical symbol,

106
00:09:53,440 --> 00:09:55,580
that is only the positive square root.

107
00:09:55,940 --> 00:10:05,240
This is only 7. 49 does have two square roots, positive and negative 7.

108
00:10:06,220 --> 00:10:10,340
But if you see the radical sign, this is only the positive.

109
00:10:10,860 --> 00:10:13,000
What is this equal to?

110
00:10:13,000 --> 00:10:34,840
And sometimes we just say rad 24. This really means the square root

111
00:10:34,840 --> 00:10:45,060
of 24. We often shorten it and say rad. Is 24 a perfect square? No. Between what

112
00:10:45,060 --> 00:10:52,960
two integers is this oh definitely between four and five it's gonna be four

113
00:10:52,960 --> 00:10:53,860
point something

114
00:10:59,820 --> 00:11:03,200
The only way to write exactly the square root of 24

115
00:11:03,200 --> 00:11:05,200
is to just leave it like this.

116
00:11:06,660 --> 00:11:08,920
However, we can simplify this.

117
00:11:11,980 --> 00:11:13,160
And here's what we do.

118
00:11:14,960 --> 00:11:17,220
What perfect square goes into 24?

119
00:11:20,100 --> 00:11:22,020
2 is not a perfect square.

120
00:11:22,020 --> 00:11:37,640
4 is a perfect square. So this factors to 4 times 6. We agree? Okay. And then one of

121
00:11:37,640 --> 00:11:44,742
our very important rules, the square root of a times b is equal to the square root

122
00:11:44,742 --> 00:11:45,160
of

123
00:11:45,160 --> 00:11:52,200
a times square root of b. Why is that true? Or is that true?

124
00:12:03,760 --> 00:12:11,817
Let's look at this. Square root of a times b is the same thing as a times b to what

125
00:12:11,817 --> 00:12:12,567
power?

126
00:12:13,840 --> 00:12:21,940
One half. And when we have a product raised to an exponent, what can I do with this?

127
00:12:23,980 --> 00:12:29,626
Raise each factor, a and b, to the one-half power: a to the one half times b to the

128
00:12:29,626 --> 00:12:30,376
one half.

129
00:12:30,960 --> 00:12:34,557
Which is one of the laws of exponents we talked about Tuesday, which is the square

130
00:12:34,557 --> 00:12:36,580
root of a times the square root of b.

131
00:12:38,420 --> 00:12:42,500
OK, so that's why this guy is true.

132
00:12:44,080 --> 00:12:45,020
So we're going to use that.

133
00:12:46,260 --> 00:12:50,420
The square root of 4 times 6 is the square root of 4 times the square root of 6.

134
00:12:52,260 --> 00:12:54,660
And the square root of 4 is 2.

135
00:12:57,000 --> 00:13:02,967
OK, so even though this cannot be written as a whole number or a decimal or

136
00:13:02,967 --> 00:13:03,717
anything,

137
00:13:04,680 --> 00:13:07,500
We still simplify it to this

138
00:13:11,460 --> 00:13:12,160
Okay

139
00:13:12,160 --> 00:13:16,100
So 50 is not a perfect square, but can we simplify it?

140
00:13:16,980 --> 00:13:18,440
What perfect square goes into 50?

141
00:13:26,760 --> 00:13:40,300
25 times 2, which I could write as rad 25 times rad 2, which is 5 rad 2.

142
00:13:41,100 --> 00:13:42,680
Five times square root of 2.

143
00:13:47,720 --> 00:13:51,621
There's a slightly easier way than trying to figure out what perfect square goes

144
00:13:51,621 --> 00:13:51,900
into

145
00:13:51,900 --> 00:13:52,800
it.

146
00:13:53,140 --> 00:13:58,292
You just take this radicand and you just start factoring it to its prime

147
00:13:58,292 --> 00:13:59,042
factorization.

148
00:14:03,420 --> 00:14:09,720
Anytime you have a pair here, it's going to be a perfect square.

149
00:14:12,760 --> 00:14:15,600
So I could write this as...

150
00:14:19,140 --> 00:14:24,880
5 squared times 2, which we usually don't write this.

151
00:14:30,920 --> 00:14:34,360
The square root of 5 squared is 5.

152
00:14:35,780 --> 00:14:40,040
So typically what I do is I factor it.

153
00:14:41,120 --> 00:14:43,860
I circle the factor pairs.

154
00:14:45,640 --> 00:14:49,480
And I know it's going to be 5, rad, whatever is left over.

155
00:14:50,800 --> 00:14:51,440
2.

156
00:14:51,440 --> 00:14:51,760
And you're done.

157
00:14:51,760 --> 00:14:54,960
it's not obvious what the perfect square goes into this

158
00:14:54,960 --> 00:14:58,300
or it might be to you, but just start factoring it. 72 is what?

159
00:14:58,500 --> 00:15:02,740
2 times 36. 2 times

160
00:15:05,960 --> 00:15:07,080
Two times

161
00:15:07,740 --> 00:15:10,580
9. 3 times 3

162
00:15:13,540 --> 00:15:17,720
Every pair is a perfect square.

163
00:15:19,460 --> 00:15:25,440
So it's going to be 2 times 3 times whatever is left over.

164
00:15:26,720 --> 00:15:27,640
Rad 2.

165
00:15:28,040 --> 00:15:29,600
So it's 6 rad 2.

166
00:15:29,600 --> 00:15:31,040
Try this one on your own.

167
00:15:33,940 --> 00:15:44,260
So, it's going to be 2 times 2 times rad 2 times 3, 4 rad 6.

168
00:15:52,860 --> 00:16:00,402
If you want to rewrite some people like this, 2 times 2 times 2 times 2 times 2

169
00:16:00,402 --> 00:16:01,240
times 3,

170
00:16:02,340 --> 00:16:09,122
Every pair is a perfect square. This becomes 2 times 2 times the square root of 2

171
00:16:09,122 --> 00:16:09,920
times 3.

172
00:16:11,840 --> 00:16:14,320
Or you can just do that part in your head.

173
00:16:22,480 --> 00:16:24,340
Is 40 a perfect cube?

174
00:16:45,320 --> 00:16:51,181
Instead of thinking what perfect squares go into this sorry perfect cubes just start

175
00:16:51,181 --> 00:16:51,931
factoring

176
00:16:52,820 --> 00:16:53,720
Two.

177
00:16:55,120 --> 00:16:56,020
Two, two.

178
00:16:57,420 --> 00:16:59,100
10 to

179
00:17:00,000 --> 00:17:00,900
5

180
00:17:02,740 --> 00:17:05,140
Now are we looking for pairs?

181
00:17:06,860 --> 00:17:14,132
Yeah, we need three of a kind because when we have three they multiply to be a

182
00:17:14,132 --> 00:17:14,560
perfect

183
00:17:14,560 --> 00:17:15,460
cube.

184
00:17:18,640 --> 00:17:24,140
So now this is going to be two times the cube root of five.

185
00:17:26,320 --> 00:17:33,180
You would not say two rad five because rad typically just means square root.

186
00:17:33,180 --> 00:17:37,320
2 times the cube root of 5.

187
00:17:37,320 --> 00:17:48,240
Just start factoring: 2 times 80; 2 times 40;

188
00:17:52,140 --> 00:17:55,300
2 times 20; 2 times 10; 2 times 5.

189
00:17:57,300 --> 00:17:59,320
Now how many do we need?

190
00:17:59,680 --> 00:18:01,720
4 We need 4 of a kind

191
00:18:04,960 --> 00:18:11,040
Which is 16 So it's going to be 2 times the 4th root of

192
00:18:11,040 --> 00:18:12,580
2 times 5.

193
00:18:12,580 --> 00:18:19,520
We're going to assume all the variables are positive.

194
00:18:21,220 --> 00:18:23,820
Now we got numbers and variables.

195
00:18:25,460 --> 00:18:26,640
Uh oh, what are we going to do there?

196
00:18:31,460 --> 00:18:32,820
Let's deal with them separately.

197
00:18:33,780 --> 00:18:36,520
So on the side, let's factor the 20.

198
00:18:42,540 --> 00:18:43,640
We've got a pair there.

199
00:18:46,320 --> 00:18:51,900
And if you factor x cubed, it's x times x times x.

200
00:18:55,800 --> 00:18:56,860
We've got a pair here.

201
00:18:57,160 --> 00:18:58,060
That's x squared.

202
00:18:59,780 --> 00:19:07,226
So, on the outside it's going to be 2 times x, and then whatever is left over, which

203
00:19:07,226 --> 00:19:07,976
is

204
00:19:08,340 --> 00:19:10,440
a 5 and an x.

205
00:19:12,040 --> 00:19:14,880
So it's 2x times rad 5x.

206
00:19:14,880 --> 00:19:16,940
Let's try another one.

207
00:19:23,680 --> 00:19:35,560
So how many x's on the outside?

208
00:19:35,560 --> 00:19:49,620
So it's going to be 3x to the power of 3 times the square root of 2x.

209
00:19:49,620 --> 00:19:55,720
Now we're going to simplify not the square roots, but cube roots.

210
00:19:56,160 --> 00:20:02,580
So, how many are we looking for at a time?

211
00:20:03,120 --> 00:20:04,240
Three of a kind.

212
00:20:08,820 --> 00:20:14,860
So we definitely have a 3 on the outside. How many x's are we looking for?

213
00:20:16,260 --> 00:20:25,940
3. So we got an x. And then we're left over with a 3 and an x squared.

214
00:20:25,940 --> 00:20:32,580
That would be it.

215
00:20:32,580 --> 00:20:34,640
Is this making sense?

216
00:20:35,380 --> 00:20:36,280
Yes.

217
00:20:36,440 --> 00:20:36,740
Okay.

218
00:20:36,740 --> 00:20:43,020
okay we're gonna write out 43 X's can circle every three of them no we need a

219
00:20:43,020 --> 00:20:47,120
So yeah, we're going to have exactly 14 x cubes

220
00:20:48,640 --> 00:20:50,680
And a remainder of one more

221
00:20:52,400 --> 00:21:00,480
So it's going to be X to the 14 times the cube root of X

222
00:21:00,480 --> 00:21:03,120
Another option is to do this

223
00:21:04,900 --> 00:21:06,660
What was the X to the 43

224
00:21:08,960 --> 00:21:12,880
Let's just change this to X to the 43 to the 1 3rd

225
00:21:16,180 --> 00:21:18,980
Which is X to the 43 times 1 3rd

226
00:21:20,940 --> 00:21:23,360
Which mixed numbers are terrible, but

227
00:21:26,060 --> 00:21:28,920
3 goes in the 43 14 times

228
00:21:31,840 --> 00:21:35,060
14 and 1 third

229
00:21:38,180 --> 00:21:42,140
But what does fourteen and one third mean?

230
00:21:44,100 --> 00:21:47,040
That means 14 plus one third.

231
00:21:49,920 --> 00:21:52,060
Mixed numbers are horrible.

232
00:21:53,980 --> 00:21:59,220
In algebra, when there is no symbol between factors, it means multiplication.

233
00:22:01,080 --> 00:22:02,500
Mixed numbers really mean addition.

234
00:22:02,500 --> 00:22:06,220
Okay, which is x to the 14th times x to the one-third.

235
00:22:07,040 --> 00:22:11,800
Which is x to the 14th times the cube root of x.

236
00:22:15,020 --> 00:22:21,470
Same idea of just dividing the exponent by the index and whatever the remainder is,

237
00:22:21,470 --> 00:22:21,900
that's

238
00:22:21,900 --> 00:22:23,100
how many are up here.

239
00:22:23,100 --> 00:22:28,820
Okay, don't start factoring 7,000.

240
00:22:30,840 --> 00:22:34,340
Is there a perfect power of 4 that goes in 7,000?

241
00:22:34,660 --> 00:22:36,060
Sorry, 70,000?

242
00:22:43,380 --> 00:22:48,520
Four zeros is ten to the power of four.

243
00:22:52,780 --> 00:23:01,060
So let's rewrite this as seven times ten to the power of four.

244
00:23:05,480 --> 00:23:08,220
And sometimes we could do well

245
00:23:12,320 --> 00:23:14,720
Four goes into four how many times

246
00:23:16,040 --> 00:23:19,720
Once okay, so this is gonna go on the outside well

247
00:23:21,200 --> 00:23:22,600
Let me do this

248
00:23:23,380 --> 00:23:27,880
We got seven out here right no we got a ten out here

249
00:23:28,740 --> 00:23:30,260
We got seven in here

250
00:23:32,140 --> 00:23:34,080
This is a perfect power of 4.

251
00:23:35,460 --> 00:23:37,420
So the 4th root of x to the 4th is an x.

252
00:23:38,900 --> 00:23:40,780
4 goes into 7 how many times?

253
00:23:42,100 --> 00:23:43,000
Once.

254
00:23:44,120 --> 00:23:45,600
And how many is left on the inside?

255
00:23:46,760 --> 00:23:47,660
3.

256
00:23:50,780 --> 00:23:52,680
4 goes into 10 twice.

257
00:23:54,940 --> 00:23:56,020
How many is left over?

258
00:23:57,440 --> 00:23:57,920
2.

259
00:23:57,920 --> 00:24:00,180
This will be z squared.