1
00:00:05,000 --> 00:00:07,680
Neither of these two terms simplify.

2
00:00:09,080 --> 00:00:11,420
Are we able to combine these together?

3
00:00:14,680 --> 00:00:17,040
Yeah, are we able to add them?

4
00:00:17,440 --> 00:00:18,340
What do they add to?

5
00:00:21,840 --> 00:00:23,400
Six square root three.

6
00:00:24,000 --> 00:00:24,700
Why is that?

7
00:00:24,700 --> 00:00:26,800
They both have a factor of square root three.

8
00:00:28,480 --> 00:00:30,320
So we could factor that out.

9
00:00:31,300 --> 00:00:35,860
You get 2 plus 4 and that is 6 square root 3.

10
00:00:38,840 --> 00:00:45,579
The radicals, the way we add them is you add the coefficients and then multiply by

11
00:00:45,579 --> 00:00:46,000
what's

12
00:00:46,000 --> 00:00:46,900
You could factor out the x.

13
00:00:47,900 --> 00:00:49,580
Ok, these are called like radicals.

14
00:00:52,080 --> 00:00:53,480
Oops.

15
00:01:00,540 --> 00:01:03,360
Three square root two. Which other term is a like radical?

16
00:01:05,640 --> 00:01:11,920
Think of this as plus negative six square root two.

17
00:01:13,860 --> 00:01:20,492
Three square root two plus negative six square root two is negative three square

18
00:01:20,492 --> 00:01:21,440
root two.

19
00:01:22,300 --> 00:01:29,340
Four square root five plus seven square root five is eleven square root five.

20
00:01:29,340 --> 00:01:38,020
you're done the directions are either simplify or it might just say add okay

21
00:01:38,020 --> 00:01:46,680
These are not like radicals yet, but both radicals simplify.

22
00:01:46,680 --> 00:01:54,280
these simplify I think so so on the side we start simplifying eight is two four

23
00:01:54,280 --> 00:01:59,300
two two this is two square root two

24
00:02:07,760 --> 00:02:10,760
And this is 5 square root 2

25
00:02:11,620 --> 00:02:16,201
Now once we simplify and we see they happen to be like radicals so we can now

26
00:02:16,201 --> 00:02:16,951
combine them

27
00:02:17,820 --> 00:02:22,800
Two square root two plus five square root two is seven square root two.

28
00:02:23,480 --> 00:02:25,820
2 plus 5 7 2

29
00:02:35,560 --> 00:02:40,560
So first we simplify all the radicals and then we combine all the like radicals.

30
00:02:41,060 --> 00:02:49,560
Makes sense? Okay. And subtracting, like I said, same thing as adding a negative so

31
00:02:49,560 --> 00:02:51,280
It's the same exact thing.

32
00:02:51,820 --> 00:02:53,100
So now we're multiplying.

33
00:02:57,540 --> 00:03:01,688
If there's more than one term, terms or things separated by addition, what property

34
00:03:01,688 --> 00:03:02,280
do we

35
00:03:02,280 --> 00:03:03,180
have to use?

36
00:03:06,980 --> 00:03:07,880
What did you just say?

37
00:03:08,360 --> 00:03:09,100
Sorry.

38
00:03:09,100 --> 00:03:14,080
So here we're multiplying, but this has more than one term.

39
00:03:15,080 --> 00:03:16,680
Terms or things separated by addition.

40
00:03:18,340 --> 00:03:23,945
What property do we have to use? Distributive. What's the long name? Did we talk

41
00:03:23,945 --> 00:03:25,440
about that last week?

42
00:03:30,680 --> 00:03:34,740
Distributive property of multiplication over addition.

43
00:03:34,740 --> 00:03:37,480
So we distribute.

44
00:03:50,740 --> 00:03:55,109
And then that's two rad, square root three times square root three is square root

45
00:03:55,109 --> 00:03:55,859
nine.

46
00:03:57,580 --> 00:03:59,700
Square root three times square root six is square root eighteen.

47
00:04:01,280 --> 00:04:03,920
and then we just start simplifying

48
00:04:07,620 --> 00:04:08,860
18 is

49
00:04:17,280 --> 00:04:20,140
Square root eighteen simplifies to three square root two.

50
00:04:23,500 --> 00:04:27,720
The result is six plus three square root two.

51
00:04:27,720 --> 00:04:29,720
We're at two plus six.

52
00:04:37,720 --> 00:04:43,940
You could first write two square root one hundred fifty, then simplify.

53
00:04:43,940 --> 00:04:51,995
at 150, but I noticed right, 5 was a factor, so I wrote it as just 5 times 5 times

54
00:04:51,995 --> 00:04:52,800
6, and

55
00:04:52,800 --> 00:04:59,063
then that's a perfect square square root 25 is 5 just kind of saves you time a

56
00:04:59,063 --> 00:04:59,813
little bit

57
00:05:02,840 --> 00:05:19,913
okay questions on that question did I screw up sorry six times to square root 10

58
00:05:19,913 --> 00:05:20,980
let's

59
00:05:20,980 --> 00:05:23,300
go 12 there we go thank you

60
00:05:23,300 --> 00:05:28,340
Now we have two terms times two terms.

61
00:05:30,360 --> 00:05:32,540
What property are we going to have to use here?

62
00:05:36,420 --> 00:05:40,620
We are still using the distributive property of multiplication.

63
00:05:42,140 --> 00:05:50,067
But instead of distributing one term to each thing separated by the addition, we

64
00:05:50,067 --> 00:05:51,200
have to

65
00:05:51,200 --> 00:05:59,340
distribute the entire thing to each term.

66
00:06:00,780 --> 00:06:09,980
Distribute the entire first binomial to each term of the second binomial.

67
00:06:19,580 --> 00:06:24,920
So you distribute both terms, or the...

68
00:06:24,920 --> 00:06:31,941
And now once you distribute both terms each term here, okay now we have to use the

69
00:06:31,941 --> 00:06:33,180
distributive property again

70
00:06:35,420 --> 00:06:37,940
Right here teach turn

71
00:06:43,480 --> 00:06:46,480
So we got two times around five

72
00:06:57,760 --> 00:07:03,980
Two terms times two terms produce four terms before combining.

73
00:07:03,980 --> 00:07:08,700
and then we combine any like terms or like radicals.

74
00:07:08,700 --> 00:07:11,760
Is that wrong?

75
00:07:24,920 --> 00:07:26,660
And are any of these like radicals?

76
00:07:27,860 --> 00:07:28,900
No. So we're done.

77
00:07:37,120 --> 00:07:40,780
Where did this subtraction come from? That sign needs to be corrected.

78
00:07:41,940 --> 00:07:45,500
We have to distribute to both terms, keeping the negative with the four.

79
00:07:51,920 --> 00:07:57,780
So yes, this right here, we have to distribute to both of these.

80
00:07:57,780 --> 00:08:05,940
thank you for picking up on that that was bad I usually just leave the

81
00:08:05,940 --> 00:08:08,720
negative with the four and then just do plus then we definitely would not have

82
00:08:08,720 --> 00:08:10,120
Here we are multiplying.

83
00:08:11,620 --> 00:08:20,340
Again, we're going to distribute this whole thing to each term.

84
00:08:34,360 --> 00:08:38,540
And now we have to use the distribute property twice.

85
00:09:00,460 --> 00:09:01,860
Oh

86
00:09:30,860 --> 00:09:34,120
And this guy simplifies right.

87
00:09:49,520 --> 00:09:53,260
And none of these are like radicals, so we're all done.

88
00:09:57,080 --> 00:10:01,340
Again this is all addition, so the order of any of those terms does not matter.

89
00:10:01,340 --> 00:10:03,760
Oh, same thing.

90
00:10:03,760 --> 00:10:08,280
I'll just review this first one.

91
00:11:09,320 --> 00:11:11,040
The middle terms add to zero, leaving two minus sixteen, or negative fourteen.

92
00:11:11,040 --> 00:11:22,860
zero zero get negative 14

93
00:11:42,860 --> 00:11:45,760
Let's try another one.

94
00:11:56,300 --> 00:11:58,440
Try this one on your own.

95
00:13:55,300 --> 00:13:57,500
The middle terms add to zero, leaving nine minus twenty-eight, or negative nineteen.

96
00:13:57,500 --> 00:14:11,640
I get 9, 7, almost 28, middle 19.

97
00:14:29,880 --> 00:14:34,500
Both conjugate products simplify to integers.

98
00:14:37,860 --> 00:14:39,140
Why is that?

99
00:14:43,780 --> 00:14:45,760
It's like a difference of squares.

100
00:14:48,500 --> 00:14:50,680
Yeah, we're going to talk about that in a minute.

101
00:14:50,680 --> 00:14:58,920
The only difference between these two sets of two terms is one has a minus one has a

102
00:14:58,920 --> 00:14:59,820
plus.

103
00:15:02,760 --> 00:15:04,100
Same thing with this one.

104
00:15:05,460 --> 00:15:08,300
Rad 2 plus 4, square root 2 minus 4.

105
00:15:10,340 --> 00:15:13,540
Whenever you have that they are called conjugates.

106
00:15:14,380 --> 00:15:22,756
When you multiply two conjugates, you always end up with the middle two terms end up

107
00:15:22,756 --> 00:15:23,280
adding

108
00:15:23,280 --> 00:15:24,180
to be zero.

109
00:15:25,800 --> 00:15:28,840
Okay, and we're going to need that later on.

110
00:15:28,840 --> 00:15:35,380
uh you know define a conjugate for you later

111
00:15:35,380 --> 00:15:39,320
That covered adding, subtracting, and multiplying. Now we are dividing radicals.

112
00:15:42,480 --> 00:15:43,640
How are we going to divide?

113
00:16:09,600 --> 00:16:10,840
Here's what we're going to do.

114
00:16:10,980 --> 00:16:19,840
This process is called rationalizing the denominator.

115
00:16:26,100 --> 00:16:27,820
What's the definition of a rational number?

116
00:16:29,100 --> 00:16:41,160
A rational number can be written as a ratio of two integers.

117
00:16:41,160 --> 00:16:44,240
a perfect square is an irrational number.

118
00:16:45,260 --> 00:16:46,480
It cannot be written as a fraction.

119
00:16:47,420 --> 00:16:52,440
So what we're going to do is we're going to rationalize the denominator.

120
00:16:52,440 --> 00:16:55,260
to turn the denominator into a rational number.

121
00:16:57,900 --> 00:16:59,540
And here's how we do it.

122
00:17:03,940 --> 00:17:05,800
We want to get rid of this square root.

123
00:17:06,860 --> 00:17:11,060
Multiply by square root two over square root two, which equals one.

124
00:17:12,700 --> 00:17:14,400
We can't just multiply by square root 2.

125
00:17:15,320 --> 00:17:17,820
But you can multiply by square root 2 over square root 2.

126
00:17:18,040 --> 00:17:18,940
Why can we do that?

127
00:17:18,940 --> 00:17:21,760
It's just number one.

128
00:17:23,120 --> 00:17:24,020
Okay.

129
00:17:24,980 --> 00:17:29,300
The denominator becomes four and the numerator becomes square root twelve.

130
00:17:31,020 --> 00:17:34,000
over two times two.

131
00:17:38,580 --> 00:17:41,520
So it's square root 12 over four.

132
00:17:44,540 --> 00:17:46,920
And then we simplify any radicals.

133
00:17:53,640 --> 00:17:54,698
Square root twelve is two square root three, so the fraction simplifies to square

134
00:17:54,698 --> 00:17:55,448
root three over two.

135
00:18:01,080 --> 00:18:06,200
And then the last thing we do is simplify.

136
00:18:07,520 --> 00:18:10,400
2 goes into the 1s.

137
00:18:10,900 --> 00:18:12,060
Rad 3 over 2.

138
00:18:15,300 --> 00:18:17,580
So we now rationalize the denominator.

139
00:18:17,580 --> 00:18:19,640
The denominator is a rational number.

140
00:18:22,160 --> 00:18:25,020
Okay, and this is basically how we're going to divide.

141
00:18:26,200 --> 00:18:30,040
All we do is rationalize the denominator

142
00:18:30,040 --> 00:18:33,180
And again, the direction might say divide or might just say simplified.

143
00:18:39,140 --> 00:18:41,260
And so what am I going to multiply up here?

144
00:18:41,260 --> 00:18:46,160
Rad 3 over square root 3.

145
00:18:48,140 --> 00:18:53,320
Multiply by square root three over square root three.

146
00:18:55,500 --> 00:18:56,980
So square root 45.

147
00:19:01,140 --> 00:19:02,400
And what is square root 45?

148
00:19:13,260 --> 00:19:15,320
Square root forty-five is three square root five, so the result is square root five.

149
00:19:16,500 --> 00:19:17,520
And this is where I left.

150
00:19:17,520 --> 00:19:19,400
Multiply by square root five over square root five.

151
00:19:19,400 --> 00:19:21,840
I'm going to do it.

152
00:19:22,180 --> 00:19:25,500
Red five by the square root five.

153
00:20:02,720 --> 00:20:09,020
The expression simplifies to two square root two over three.

154
00:20:19,580 --> 00:20:21,060
We've got questions on that.

155
00:20:22,220 --> 00:20:23,820
Again, we rationalized the denominator.

156
00:20:23,980 --> 00:20:25,280
This is not a rational number.

157
00:20:25,960 --> 00:20:28,200
So no, we don't want any square roots in the denominator.

158
00:20:30,440 --> 00:20:31,880
Let's try another one.

159
00:20:31,880 --> 00:20:33,340
Do not multiply this two-term denominator by square root five alone.

160
00:20:34,120 --> 00:20:35,780
We're not going to multiply by square root 5.

161
00:20:37,760 --> 00:20:44,860
Because if you do, we get 3 square root 5 plus 5.

162
00:20:45,760 --> 00:20:47,380
We did get rid of this square root.

163
00:20:49,080 --> 00:20:51,080
But now we got another one stuck here.

164
00:20:54,760 --> 00:20:56,120
Multiply by square root 5 again.

165
00:20:56,900 --> 00:20:58,600
Yeah, you should try that.

166
00:20:58,740 --> 00:20:59,640
Try that.

167
00:21:02,660 --> 00:21:10,060
Now we're at 2 times 5 is 10 over 3 times 5 plus 5 right 5.

168
00:21:10,620 --> 00:21:11,500
Wait, one more time.

169
00:21:11,500 --> 00:21:12,400
One more time?

170
00:21:12,500 --> 00:21:13,400
Yeah.

171
00:21:15,260 --> 00:21:16,300
Okay, so that doesn't work.

172
00:21:24,760 --> 00:21:26,760
What did somebody say over there?

173
00:21:28,160 --> 00:21:32,900
Multiply by the conjugate of the denominator.

174
00:21:36,100 --> 00:21:43,980
The conjugate of three plus square root five is three minus square root five.

175
00:21:43,980 --> 00:21:50,500
So we're going to multiply by 3 minus square root 5 or 3 minus square root 5.

176
00:21:58,280 --> 00:22:00,480
Now we've got a lot of distributing.

177
00:22:06,400 --> 00:22:09,440
6 minus 2 square root 5.

178
00:23:00,460 --> 00:23:06,600
And once you multiply by the conjugate, the two middle terms always has to be zero.

179
00:23:11,940 --> 00:23:18,980
After multiplying by the conjugate, the two middle denominator terms cancel.

180
00:23:22,380 --> 00:23:23,960
And we're almost done.

181
00:23:25,160 --> 00:23:26,320
Why are we not quite done?

182
00:23:27,740 --> 00:23:30,160
We definitely rationalized the denominator.

183
00:23:31,680 --> 00:23:35,340
But they all have a factor of 2 in it.

184
00:23:38,100 --> 00:23:39,560
Okay, you could do this.

185
00:23:39,740 --> 00:23:43,140
6 over 4 minus 2 square root 5 over 4.

186
00:23:44,420 --> 00:23:46,460
Which is 3 over 2 minus

187
00:23:48,300 --> 00:23:50,300
square root 5 over 2.

188
00:23:52,900 --> 00:23:57,540
Factor two from the numerator and simplify to three minus square root five over two.

189
00:24:00,700 --> 00:24:07,280
The other option is you just factor out a 2.

190
00:24:07,280 --> 00:24:16,500
We get 2 times 3 minus square root phi over 4.

191
00:24:19,400 --> 00:24:20,920
And that simplifies the one.

192
00:24:20,920 --> 00:24:31,380
okay questions on that so if we're dividing by something with two terms as

193
00:24:31,380 --> 00:24:36,499
As a radical we have to multiply by the conjugate over the conjugate of the

194
00:24:36,499 --> 00:24:37,249
denominator

195
00:24:38,820 --> 00:24:40,880
Let's try this one

196
00:25:12,880 --> 00:25:16,460
Are we going to multiply by the conjugate of the numerator or denominator?

197
00:25:17,600 --> 00:25:18,160
Denominator.

198
00:25:18,160 --> 00:25:20,320
Our whole goal is to get rid of square roots in the denominator.

199
00:25:21,660 --> 00:25:24,120
So we're going to multiply by 3.

200
00:25:24,460 --> 00:25:25,540
You just switch the sign.

201
00:25:25,540 --> 00:25:37,120
plus square root 3 plus square root 7. And now we have lots of distributing to do.

202
00:26:48,800 --> 00:26:50,280
Distribute carefully through the numerator and denominator.

203
00:26:50,300 --> 00:26:51,800
Don't mess up.

204
00:26:55,540 --> 00:27:03,340
plus four and six, plus seven, plus 14.

205
00:27:10,420 --> 00:27:12,880
It becomes three.

206
00:27:48,260 --> 00:27:51,140
Are any of the ones in the numerator like radicals?

207
00:27:55,140 --> 00:28:09,620
no that's gonna be it right there it's the best we could do okay what

208
00:28:09,620 --> 00:28:13,520
Adding and subtracting use like radicals; multiplying uses distribution; dividing

209
00:28:13,520 --> 00:28:15,080
requires rationalizing the denominator.

210
00:28:15,080 --> 00:28:19,900
dividing divided is the only weird one we don't really divide we multiply by

211
00:28:19,900 --> 00:28:27,120
something to rationalize the denominator when there's two terms you multiply by

212
00:28:27,120 --> 00:28:37,400
the conjugate so a plus B the conjugate

213
00:28:40,080 --> 00:28:47,580
is A minus B and vice versa. These are conjugates. All we do is switch the term.