1 00:00:00,750 --> 00:00:04,655 Today we're going to look at numbers, written his powers, and 2 00:00:04,655 --> 00:00:08,205 we're going to go through some calculations involving them, and 3 00:00:08,205 --> 00:00:11,400 in particular, we're going to look at square roots. 4 00:00:12,180 --> 00:00:16,448 And the square roots of numbers, which give us an irrational 5 00:00:16,448 --> 00:00:18,250 answer. That is. 6 00:00:18,800 --> 00:00:23,123 Numbers which can be written as whole numbers are fractions. Now 7 00:00:23,123 --> 00:00:25,481 these types of square roots are 8 00:00:25,481 --> 00:00:28,068 called Surds. But more of that 9 00:00:28,068 --> 00:00:31,830 later. What I want to do first is a little bit of revision. 10 00:00:32,710 --> 00:00:35,718 And we'll start with something that we know. 11 00:00:36,810 --> 00:00:43,506 2 cubed Is written as 2 * 2 * 2. 12 00:00:45,000 --> 00:00:48,689 It's a neat way of writing this 13 00:00:48,689 --> 00:00:55,052 repeated multiplication. And we say that three is the power or 14 00:00:55,052 --> 00:01:00,233 index. And two is rears to that power or index. 15 00:01:01,720 --> 00:01:06,616 And the value of 2 cubed works out to be it because 16 00:01:06,616 --> 00:01:09,064 it's 2 * 2 * 2. 17 00:01:10,450 --> 00:01:14,410 So if we had 4 cubed. 18 00:01:15,430 --> 00:01:19,350 That is 4 * 4 19 00:01:19,350 --> 00:01:24,120 * 4. And we know when that's worth. Diet is 64. 20 00:01:24,840 --> 00:01:27,512 But what if we have 4 to the 21 00:01:27,512 --> 00:01:32,753 negative 3? What would be its value and high? Could we put it 22 00:01:32,753 --> 00:01:33,782 down more deeply? 23 00:01:34,400 --> 00:01:36,374 Well, we go back to what we 24 00:01:36,374 --> 00:01:42,365 know. 4 cubed equals 4 * 4 * 4, which is 64. 25 00:01:43,220 --> 00:01:49,788 4 squared equals 4 * 4 and not 26 00:01:49,788 --> 00:01:56,860 equals 16. 14 power one is for. 27 00:01:56,860 --> 00:01:58,520 Is for. 28 00:01:59,640 --> 00:02:05,451 An forwarded the zero. Well, if we look at our pattern in our 29 00:02:05,451 --> 00:02:09,974 calculations. We are dividing by for each time. 30 00:02:10,570 --> 00:02:16,785 And the index. Our power decreases by one. So forward the 31 00:02:16,785 --> 00:02:20,740 zero must be 4 / 4 which 32 00:02:20,740 --> 00:02:27,220 equals 1. But we were interested in finding out about four to the 33 00:02:27,220 --> 00:02:30,820 negative three, so will continue our pattern of multiplications. 34 00:02:31,330 --> 00:02:38,974 4 to the negative one will be the 1 / 4 which 35 00:02:38,974 --> 00:02:45,660 is 1/4. And forward to the negative two must be 1/4. 36 00:02:46,220 --> 00:02:49,790 Divided by 4, which is 16. 37 00:02:50,610 --> 00:02:57,762 An four to the negative three must be 116th divided by 4, 38 00:02:57,762 --> 00:03:00,742 which is one over 64. 39 00:03:02,050 --> 00:03:09,086 But If we go back to this line, we knew that 4 cubed is 40 00:03:09,086 --> 00:03:14,832 64, so we can rewrite one over 64 as one over 4 cubed. 41 00:03:15,690 --> 00:03:19,516 And look at that. Forward to the negative three is 42 00:03:19,516 --> 00:03:21,020 one over 4 cubed. 43 00:03:23,060 --> 00:03:29,948 So for the negative two is one over 16 must be one over 4 44 00:03:29,948 --> 00:03:34,956 squared. And four to the negative, one must equal 45 00:03:34,956 --> 00:03:39,873 for one over 4th power, one which is one over 4. 46 00:03:41,630 --> 00:03:47,140 So negative powers give you the reciprocal of the number. 47 00:03:47,810 --> 00:03:50,456 That is one over the number. 48 00:03:51,310 --> 00:03:55,066 So if I had this example. 49 00:03:56,650 --> 00:03:59,170 3 to the negative 2. 50 00:03:59,760 --> 00:04:04,170 We can write that as one over 3 squared. 51 00:04:06,050 --> 00:04:09,782 And one over 3 squared is the same as one over 9. 52 00:04:11,260 --> 00:04:16,512 What about this one? 5 to the negative 3? How can we rewrite 53 00:04:16,512 --> 00:04:21,360 that and what's the value? Well, the negative 3 means it's one 54 00:04:21,360 --> 00:04:24,270 over. 5 cubed 55 00:04:25,150 --> 00:04:32,206 And that is one over 5 cubed is 125. So 5 to the negative 56 00:04:32,206 --> 00:04:34,726 three is one over 125. 57 00:04:36,950 --> 00:04:41,929 Not the big thing to note is that even though you've got a 58 00:04:41,929 --> 00:04:46,142 negative power, that value that you get is a positive answer. 59 00:04:46,142 --> 00:04:51,121 That's a big mistake by quite a lot of people. They think when 60 00:04:51,121 --> 00:04:55,334 you've got a negative power, you'll get a negative answer, so 61 00:04:55,334 --> 00:04:58,781 please don't make that mistake. And negative power means 62 00:04:58,781 --> 00:05:01,845 reciprocal of the number one over the power. 63 00:05:03,020 --> 00:05:08,924 Moving on in the examples that I've just been doing, we've used 64 00:05:08,924 --> 00:05:14,336 positive negative integers for the powers, but what if we have 65 00:05:14,336 --> 00:05:16,304 fractional indices or fractional 66 00:05:16,304 --> 00:05:20,106 powers? How can we represent them? What's their value? 67 00:05:20,750 --> 00:05:22,112 Well again will start with what 68 00:05:22,112 --> 00:05:27,846 we know. We know that force of power one is 4. 69 00:05:28,460 --> 00:05:33,685 But what would be for to the half? What's it value? 70 00:05:34,190 --> 00:05:37,775 We're looking at this, we 71 00:05:37,775 --> 00:05:44,326 know using. Indices laws, therefore, to the half times by 72 00:05:44,326 --> 00:05:48,015 4 to the half, must equal 4. 73 00:05:48,700 --> 00:05:50,910 To the indices added together. 74 00:05:51,500 --> 00:05:53,450 And 1/2 + 1/2 is one. 75 00:05:54,460 --> 00:05:58,324 Before to the one is just 4. 76 00:05:59,420 --> 00:06:03,430 So 4 to the half times, 4 1/2 equals 4. 77 00:06:04,790 --> 00:06:08,860 That means that it's something times by itself equals 4. Well, 78 00:06:08,860 --> 00:06:13,300 we know that it's obvious it's two 2 * 2 equals 4. 79 00:06:14,250 --> 00:06:17,267 So 4 to the half equals 2. 80 00:06:17,780 --> 00:06:21,280 So. Forza half. 81 00:06:21,880 --> 00:06:23,120 Is equal to 2. 82 00:06:23,870 --> 00:06:28,719 And we write this in a shorthand notation by doing this for the 83 00:06:28,719 --> 00:06:30,957 half is equal to the square 84 00:06:30,957 --> 00:06:36,423 root. Of four, and that is the symbol for the square root. 85 00:06:36,930 --> 00:06:41,583 So the half is associated with the square routing of a 86 00:06:41,583 --> 00:06:47,505 number. So if we had 9 to the power of 1/2, that's the same 87 00:06:47,505 --> 00:06:51,312 as the square root of 9, which equals 3. 88 00:06:52,440 --> 00:06:58,251 So in general, if we had a number a razor, the part half. 89 00:06:58,810 --> 00:07:02,550 Then we can write that as the square root of A. 90 00:07:03,860 --> 00:07:08,804 We can continue this on by saying if we wanted the cube 91 00:07:08,804 --> 00:07:11,276 root of a number, say for. 92 00:07:12,260 --> 00:07:18,045 To the third, we can write that as the cube root of 4. 93 00:07:19,440 --> 00:07:20,230 Like this? 94 00:07:21,340 --> 00:07:26,873 And the fraction associated with the cube root is 1/3. Similarly, 95 00:07:26,873 --> 00:07:33,915 the 4th root. So if we had five, the 4th root of 5, you 96 00:07:33,915 --> 00:07:38,945 can write that using this shorthand notation with a small 97 00:07:38,945 --> 00:07:44,478 for their and then this root symbol. And the quarter is 98 00:07:44,478 --> 00:07:46,993 associated with the 4th root. 99 00:07:48,390 --> 00:07:50,616 So if we had the NTH root. 100 00:07:51,170 --> 00:07:53,858 Of a number A. 101 00:07:54,380 --> 00:07:59,076 We can write it like this in shorthand. 102 00:08:01,940 --> 00:08:05,504 So if we have the cube root of 64. 103 00:08:06,730 --> 00:08:09,796 We can write it like this. 104 00:08:10,540 --> 00:08:17,580 And it equals 64. Raise the part of 1/3. Now 105 00:08:17,580 --> 00:08:23,916 we know 64 is 4 * 4 * 4. 106 00:08:24,450 --> 00:08:28,206 And that's raised to the power 107 00:08:28,206 --> 00:08:33,112 of 1/3. And the answer is dead easy. We want to know what 108 00:08:33,112 --> 00:08:36,764 number times by itself three times gives you 64. Well, it's 109 00:08:36,764 --> 00:08:40,748 obvious because what we've just written it must be for. So the 110 00:08:40,748 --> 00:08:42,740 cube root of 64 is 4. 111 00:08:44,920 --> 00:08:50,240 A very useful simple fact leading on from this that. 112 00:08:51,100 --> 00:08:56,651 Is known, but people tend to forget it is that we know that 113 00:08:56,651 --> 00:09:01,775 forward to the half times by 4 to the half equals 4. 114 00:09:02,890 --> 00:09:09,580 That is the square root of 4 times by the square root of 4 is 115 00:09:09,580 --> 00:09:15,378 4 and we can write that more simply by saying is the square 116 00:09:15,378 --> 00:09:18,054 root of 4 squared equals 4. 117 00:09:19,240 --> 00:09:24,268 Not, I'll be using this very simple fact. Little role at the 118 00:09:24,268 --> 00:09:29,296 end of the session, so remember it. It's any number. The square 119 00:09:29,296 --> 00:09:31,810 root of it squared gives you 120 00:09:31,810 --> 00:09:37,411 that number. Another important thing to remember about serves 121 00:09:37,411 --> 00:09:43,441 comes from solving this equation. If we had this very 122 00:09:43,441 --> 00:09:47,662 simple quadratic equation X squared equal 4. 123 00:09:48,460 --> 00:09:54,312 If we take the square root of four, we know that we have to 124 00:09:54,312 --> 00:09:55,566 have two routes. 125 00:09:56,100 --> 00:10:01,039 For the solution, because this is a quadratic equation, so X 126 00:10:01,039 --> 00:10:05,080 equals plus or minus the square root of 4. 127 00:10:05,640 --> 00:10:09,464 So we know that is positive two or 128 00:10:09,464 --> 00:10:12,332 negative two. You've got 2 routes. 129 00:10:13,710 --> 00:10:17,600 So. Roots are not unique. 130 00:10:18,250 --> 00:10:22,354 And remember, we don't always have to write the 131 00:10:22,354 --> 00:10:26,458 positive side. We can write positive two as two. 132 00:10:27,650 --> 00:10:32,824 And remember. If you haven't got a sign in front of the square 133 00:10:32,824 --> 00:10:34,708 root, then it's assumed it's a 134 00:10:34,708 --> 00:10:39,370 positive square root. When it's not, you have to put the sign in 135 00:10:39,370 --> 00:10:41,032 and make it a negative square 136 00:10:41,032 --> 00:10:44,628 root. But the positive square roots of the ones 137 00:10:44,628 --> 00:10:46,083 that are are most common. 138 00:10:48,300 --> 00:10:49,719 Now moving on. 139 00:10:50,280 --> 00:10:52,268 What I'd like to do is actually 140 00:10:52,268 --> 00:10:54,600 do some work. On 141 00:10:55,780 --> 00:10:57,980 The square root of negative 142 00:10:57,980 --> 00:11:03,059 numbers. We've done the square root of positive numbers. What 143 00:11:03,059 --> 00:11:06,076 if we had the square root of 144 00:11:06,076 --> 00:11:09,125 negative 9? Can we evaluate 145 00:11:09,125 --> 00:11:13,987 this? Well, the square root of negative 9 can be written as 146 00:11:13,987 --> 00:11:15,889 negative 9 raised to the power 147 00:11:15,889 --> 00:11:21,280 of 1/2. And So what we're looking for are two numbers 148 00:11:21,280 --> 00:11:25,480 multiplied by themselves, which will give us negative. Now when 149 00:11:25,480 --> 00:11:28,420 we know 3 * 3 is 9. 150 00:11:29,270 --> 00:11:33,166 And negative three times negative three is 9. 151 00:11:34,410 --> 00:11:35,430 But there's not. 152 00:11:36,120 --> 00:11:39,405 2 numbers multiply by themselves, which will give us 153 00:11:39,405 --> 00:11:42,658 negative night. So you cannot evaluate the square 154 00:11:42,658 --> 00:11:45,838 root of a negative number and get a real answer. 155 00:11:48,200 --> 00:11:55,376 Now what I'd like to do is continue on using surds in some 156 00:11:55,376 --> 00:12:00,407 calculations. And go over some common square roots that you 157 00:12:00,407 --> 00:12:07,934 might know. Now we all know that the square root of 25 is 5 158 00:12:07,934 --> 00:12:10,856 because 5 * 5 is 25. 159 00:12:11,470 --> 00:12:15,910 And if we were given the square root of 9 over 4, 160 00:12:15,910 --> 00:12:21,090 that's dead easy. It must be 3 over 2 because 3 * 3 is 161 00:12:21,090 --> 00:12:26,640 nine, 2 * 2 is 4, and we write that as one and a half. 162 00:12:28,540 --> 00:12:32,357 But we've now taken square root of whole numbers and fractions, 163 00:12:32,357 --> 00:12:35,827 and we've got answers that are whole numbers and fractions. 164 00:12:36,360 --> 00:12:41,616 But there are other square roots in which we don't get whole 165 00:12:41,616 --> 00:12:45,558 numbers or fractions. We get them as non terminating 166 00:12:45,558 --> 00:12:48,878 decimals. Such as Route 2. 167 00:12:49,410 --> 00:12:55,638 Route 2 cannot be evaluated as a whole number or a fraction. 168 00:12:55,638 --> 00:12:57,714 Never can Route 3. 169 00:12:59,240 --> 00:13:04,104 We can only write them as approximations. They 170 00:13:04,104 --> 00:13:05,928 are irrational numbers. 171 00:13:07,040 --> 00:13:13,340 An approximation for the square root of 2 to 3 decimal places 172 00:13:13,340 --> 00:13:19,696 is 1.414. Approximation from Route 3 to 3 decimal places, 173 00:13:19,696 --> 00:13:26,860 and I put 3 decimal places just to remind us is 1.732. 174 00:13:27,780 --> 00:13:33,104 Not because. They cannot be evaluated as whole numbers 175 00:13:33,104 --> 00:13:36,338 and fractions, and these are irrational numbers. 176 00:13:37,910 --> 00:13:41,969 They are very special and we call them surged. 177 00:13:42,550 --> 00:13:50,314 Other common starts would be Route 178 00:13:50,314 --> 00:13:57,988 5. Rich sex, which 7 Route 8 not Route 9 because we 179 00:13:57,988 --> 00:14:01,396 know that's three and Route 10. 180 00:14:02,730 --> 00:14:04,277 Now, why do we need to know 181 00:14:04,277 --> 00:14:08,935 about search? Will search crop up a lot when we using 182 00:14:08,935 --> 00:14:12,985 Pythagoras Theorem and in trigonometry and so we need to 183 00:14:12,985 --> 00:14:15,415 know how to manipulate these in 184 00:14:15,415 --> 00:14:19,356 various calculations. And we need to know how to simplify 185 00:14:19,356 --> 00:14:20,187 them as well. 186 00:14:21,360 --> 00:14:26,520 Now Route 2 cannot be simplified any further. Now they can Route 187 00:14:26,520 --> 00:14:33,400 3 or Route 5 with six, 3, Seven, but look at root it. If we take 188 00:14:33,400 --> 00:14:35,550 the square root of it. 189 00:14:37,610 --> 00:14:42,380 It can be replaced by the product 4 * 2. 190 00:14:43,300 --> 00:14:46,786 Now the square root of product. 191 00:14:47,600 --> 00:14:52,696 Is the product of the square roots, so it's Square root of 4 192 00:14:52,696 --> 00:14:55,440 times by the square root of 2. 193 00:14:56,580 --> 00:14:59,444 We know the square root of 4 is 194 00:14:59,444 --> 00:15:05,330 2. And this is Route 2. So the square root of it can be written 195 00:15:05,330 --> 00:15:09,590 more simply as two square root of two or two Route 2. 196 00:15:10,160 --> 00:15:16,438 In shorthand. What about this one? What about the square root 197 00:15:16,438 --> 00:15:19,952 of 5 times by the square root 198 00:15:19,952 --> 00:15:25,680 of 15? Now there in the previous example, the product of the 199 00:15:25,680 --> 00:15:31,344 square roots is equal to the square root of the products, so 200 00:15:31,344 --> 00:15:33,704 we use that fact too. 201 00:15:34,270 --> 00:15:36,274 Rewrite this expression in 202 00:15:36,274 --> 00:15:42,082 another way. The square root of 5 times by the square root of 15 203 00:15:42,082 --> 00:15:44,890 is the square root of 5 times by 204 00:15:44,890 --> 00:15:49,826 15. And that is the square root of 75. 205 00:15:50,450 --> 00:15:58,260 But 75 can be rewritten as 25 times by three. 206 00:15:59,610 --> 00:16:05,420 Note 25 is a square number and we can then separate that out to 207 00:16:05,420 --> 00:16:11,645 say that is the square root of 25 times by the square root of 3 208 00:16:11,645 --> 00:16:18,285 square root of 25 is 5 times by Route 3. So the square root of 5 209 00:16:18,285 --> 00:16:24,095 times the square root of 15 is 5 route 3, but could be easier. 210 00:16:24,970 --> 00:16:31,700 But a common mistake is the people would take an 211 00:16:31,700 --> 00:16:33,719 expression like this. 212 00:16:34,560 --> 00:16:39,248 I served in the form of the square root of 4 + 9 and say is 213 00:16:39,248 --> 00:16:43,057 equal to the square root of 4 plus the square root of 9. 214 00:16:43,640 --> 00:16:47,024 Now this can't happen because if we follow this 215 00:16:47,024 --> 00:16:51,160 through that would be 2 + 3, which would equal 5. 216 00:16:52,450 --> 00:16:57,106 This is not correct because five is the square root of 25. 217 00:16:57,770 --> 00:17:05,370 And here we've got 4 + 9, which is 13. So the square root of an 218 00:17:05,370 --> 00:17:10,595 addition is not equal to the square root plus the other 219 00:17:10,595 --> 00:17:12,970 square root. So Please remember 220 00:17:12,970 --> 00:17:18,970 that. Those two rules that we previously did with Surds, an 221 00:17:18,970 --> 00:17:22,386 expansion of multiplications only applies to multiplication 222 00:17:22,386 --> 00:17:25,802 and division, not to addition and Subtraction. 223 00:17:29,230 --> 00:17:31,250 So if we move on. 224 00:17:32,000 --> 00:17:38,325 And we take the square root of bigger numbers times by. 225 00:17:38,330 --> 00:17:41,347 Square root of a another big number. 226 00:17:42,630 --> 00:17:46,410 The square root of 400 times by the square root of 90. 227 00:17:47,480 --> 00:17:50,959 It's useful to remember square numbers. The 228 00:17:50,959 --> 00:17:52,450 typical square numbers. 229 00:17:53,720 --> 00:18:00,056 Square numbers of the numbers 1 to 10 at one, four 916-2549, 230 00:18:00,056 --> 00:18:05,864 sixty, 481 and 100. You should be able to remember those 231 00:18:05,864 --> 00:18:11,672 straight off and recall them, and this helps us to rewrite 232 00:18:11,672 --> 00:18:17,986 these. Square roots we can write square root of 400 as the square 233 00:18:17,986 --> 00:18:20,614 root of 4 times by 100. 234 00:18:21,310 --> 00:18:26,238 And the square root of 90 is the square root of 9 * 10. 235 00:18:26,750 --> 00:18:29,370 Then we expand out. 236 00:18:30,080 --> 00:18:34,877 And we say that this product and the square root of that product 237 00:18:34,877 --> 00:18:39,674 is the square root of 4 times by the square root of 100. 238 00:18:40,220 --> 00:18:45,232 Times by the square root of 9 times by the square root of 10. 239 00:18:45,790 --> 00:18:50,230 Then we simplify. We know the square root of 4 because there's 240 00:18:50,230 --> 00:18:55,410 a square number and it's two. We know the square root of 100 is 241 00:18:55,410 --> 00:18:58,000 10 square root of 9 is 3. 242 00:18:58,600 --> 00:19:02,799 Thought the square root of 10 is just the square root of 10. 243 00:19:04,450 --> 00:19:08,970 And we meeting it up by multiplying these three numbers 244 00:19:08,970 --> 00:19:14,394 together. That's 2 * 10 * 3 is 6060 Route 10 delicious. 245 00:19:15,000 --> 00:19:18,066 What if we had a quotient? 246 00:19:18,970 --> 00:19:20,728 Involving square roots. 247 00:19:21,290 --> 00:19:26,217 What if we had the square root of 2000 divided by the square 248 00:19:26,217 --> 00:19:31,523 root of 50? Well, we use a similar rule to the way that we 249 00:19:31,523 --> 00:19:36,450 use the rule for multiplication. We can say that that is equal to 250 00:19:36,450 --> 00:19:39,103 the square root of 2000 / 50. 251 00:19:39,640 --> 00:19:42,148 We cancelled I'm. 252 00:19:42,760 --> 00:19:44,470 And we get. 253 00:19:45,300 --> 00:19:51,680 That The answer is the square root of 40, but we can simplify 254 00:19:51,680 --> 00:19:56,245 the square root of 40. Thinking of square numbers breakdown 40 255 00:19:56,245 --> 00:20:02,055 as 4 * 10 and that equals the square root of four times the 256 00:20:02,055 --> 00:20:07,450 square root of 10, which is 2 times the square root of 10. 257 00:20:07,450 --> 00:20:11,185 What could be simpler? It becomes second nature really, 258 00:20:11,185 --> 00:20:15,750 the more practice you do with the rules, the more you 259 00:20:15,750 --> 00:20:16,995 understand them and. 260 00:20:17,020 --> 00:20:18,266 You don't have to think about them. 261 00:20:19,850 --> 00:20:25,250 Now, these simplifications of Surds are quite easy ones. 262 00:20:25,990 --> 00:20:30,220 But what if we had this expression involving Surds? 263 00:20:30,760 --> 00:20:36,250 It's a little bit different from the expressions that we've 264 00:20:36,250 --> 00:20:41,820 already used. One plus square root of 3 times by two minus the 265 00:20:41,820 --> 00:20:46,140 square root of 2. How do we expand out an expression like 266 00:20:46,140 --> 00:20:51,180 this where we do it just in the same way as we did normally? 267 00:20:52,210 --> 00:20:56,722 We use the first number in these brackets, multiply it by these 268 00:20:56,722 --> 00:21:01,234 two numbers, and then we use this number and multiply it by 269 00:21:01,234 --> 00:21:05,370 these two numbers and combine them together if possible. So if 270 00:21:05,370 --> 00:21:10,634 we do that, we multiply 1 by these two numbers and we get 2 271 00:21:10,634 --> 00:21:15,522 minus the square root of 2 and we multiplied by square root of 272 00:21:15,522 --> 00:21:16,650 3 that is. 273 00:21:17,410 --> 00:21:24,162 Square root of 3 * 2 is 2 square root of 3 and then the square 274 00:21:24,162 --> 00:21:29,226 root of 3 times negative square root of 2, which is your 275 00:21:29,226 --> 00:21:32,708 subtract. Route 3 Route 2. 276 00:21:33,710 --> 00:21:35,558 Now when we look at that 277 00:21:35,558 --> 00:21:39,900 expression. And consider it you can see that we can't really 278 00:21:39,900 --> 00:21:41,308 simplify it any further. 279 00:21:42,880 --> 00:21:46,408 But what if we had this 280 00:21:46,408 --> 00:21:51,516 expression? Similar to the one above using two brackets, but 281 00:21:51,516 --> 00:21:56,400 look at the numbers and the 3rd's involved. One plus square 282 00:21:56,400 --> 00:22:02,616 root of 3 times by one minus the square root of 3. The numbers 283 00:22:02,616 --> 00:22:07,056 the same, but the operations different. See what happens when 284 00:22:07,056 --> 00:22:13,716 we multiply out in the usual way. 1 * 1 minus Route 3 is 1 285 00:22:13,716 --> 00:22:19,488 minus Route 3, then Route 3 times these two Route 3 * 1. 286 00:22:19,490 --> 00:22:26,990 Is plus 3 route 3 times by minus 3 three is subtract. 287 00:22:26,990 --> 00:22:29,490 Route 3 route 3. 288 00:22:30,200 --> 00:22:36,111 Now remember. That little fact that we did previously. This is 289 00:22:36,111 --> 00:22:42,208 Route 3 squared and what's Route 3 squared? It's just three so we 290 00:22:42,208 --> 00:22:48,305 can look at our expression and we can see that that is one. 291 00:22:48,870 --> 00:22:51,390 And loan behold, look what happens here in the middle. 292 00:22:52,670 --> 00:22:55,718 Subtract right, three, add Route 3, they cancel. 293 00:22:56,300 --> 00:22:59,324 And so we're left with one 294 00:22:59,324 --> 00:23:06,410 minus. This which is three 1 - 3 is negative 2. 295 00:23:08,380 --> 00:23:12,349 So we started off with a multiplication of brackets, 296 00:23:12,349 --> 00:23:16,759 which involves surds. But look what happens with the answer. 297 00:23:16,759 --> 00:23:20,728 We've got a whole number with no service involved. 298 00:23:22,130 --> 00:23:27,109 Now this is a significant fact, and in fact this is so well 299 00:23:27,109 --> 00:23:31,705 known it's given a name. This type of expansion is given a 300 00:23:31,705 --> 00:23:34,769 name. It's called the difference of two squares. 301 00:23:35,390 --> 00:23:41,418 And the difference of two squares is written as a squared 302 00:23:41,418 --> 00:23:45,254 minus B squared, and it can be 303 00:23:45,254 --> 00:23:47,448 expanded. By using two brackets. 304 00:23:48,630 --> 00:23:52,402 In two A-B times 305 00:23:52,402 --> 00:23:58,908 by A+B? Now, can you see that this is similar to 306 00:23:58,908 --> 00:24:04,676 what we had up here, where a is one and B is Route 3? 307 00:24:05,840 --> 00:24:13,540 So. Our product here. One plus Route 3 times by one 308 00:24:13,540 --> 00:24:20,437 minus Route 3 is really the difference of two squares and 309 00:24:20,437 --> 00:24:24,199 the two squares are 1 squared. 310 00:24:24,870 --> 00:24:28,430 Minus Route 3 squared. 311 00:24:29,360 --> 00:24:34,920 Again, notice we're using that simple fact, which makes things 312 00:24:34,920 --> 00:24:41,036 really easy. That is 1 - 3, which is negative 2. 313 00:24:42,050 --> 00:24:44,475 So if you remember this 314 00:24:44,475 --> 00:24:49,680 expansion. The difference of two squares. This is very 315 00:24:49,680 --> 00:24:53,355 helpful sometimes when you have calculations involving 316 00:24:53,355 --> 00:24:54,405 sired serves. 317 00:24:56,850 --> 00:25:03,527 Night, sometimes you have surged written like this one over the 318 00:25:03,527 --> 00:25:05,955 square root of 13. 319 00:25:06,930 --> 00:25:11,350 Now really, we don't like answers being given such that 320 00:25:11,350 --> 00:25:16,654 the denominator is assert. What we want to do is give an 321 00:25:16,654 --> 00:25:20,190 answer with a whole number as the denominator. 322 00:25:21,250 --> 00:25:25,678 Now there is a technique that helps us to change this sired 323 00:25:25,678 --> 00:25:28,630 into whole number and that technique is called 324 00:25:28,630 --> 00:25:32,689 rationalization. What we have to do is rationalize the Route 30 325 00:25:32,689 --> 00:25:38,224 we have to make it into a whole number. Now the easy way to do 326 00:25:38,224 --> 00:25:42,652 that is just use that simple fact that keeps cropping up time 327 00:25:42,652 --> 00:25:47,080 and time again is that we have to multiply a thigh itself. 328 00:25:47,790 --> 00:25:52,743 But we don't want to change the value of this fraction. So what 329 00:25:52,743 --> 00:25:55,791 we have to do is multiply it by. 330 00:25:56,360 --> 00:26:00,920 The fraction square root of 13 over the square root of 13 331 00:26:00,920 --> 00:26:02,820 because that fraction is equal 332 00:26:02,820 --> 00:26:06,542 to 1. Now when we do 333 00:26:06,542 --> 00:26:08,980 that. And evaluated. 334 00:26:09,570 --> 00:26:14,442 The square root of 13 times the square root of 13 is there Ting 335 00:26:14,442 --> 00:26:19,662 and on top one times the square root of 13 is the square root of 336 00:26:19,662 --> 00:26:25,080 30. And this is the more acceptable form of writing one 337 00:26:25,080 --> 00:26:27,714 over the square root of 30. 338 00:26:28,420 --> 00:26:32,105 It's where you have, as I said before, a whole number 339 00:26:32,105 --> 00:26:33,110 as the denominator. 340 00:26:34,470 --> 00:26:39,510 Now we can get more complicated fractions, which involves 341 00:26:39,510 --> 00:26:46,230 serves. What if we had this fraction one over 1 plus the 342 00:26:46,230 --> 00:26:48,470 square root of 2? 343 00:26:49,540 --> 00:26:54,190 Hi can be rationalized that high can we make this? 344 00:26:54,760 --> 00:26:57,310 Addition into a whole number. 345 00:26:58,340 --> 00:27:03,170 Will remember what we've just done on the previous examples. 346 00:27:03,690 --> 00:27:05,778 We use the difference of two 347 00:27:05,778 --> 00:27:09,884 squares. And the difference two squares had an expansion where 348 00:27:09,884 --> 00:27:13,360 you had the two numbers the same, but the operation was 349 00:27:13,360 --> 00:27:18,487 different. So what if we multiplied the one plus the 350 00:27:18,487 --> 00:27:22,111 square root of 2 by 1 minus root 351 00:27:22,111 --> 00:27:26,733 2? Now what we've got on the bottom is the expansion of 352 00:27:26,733 --> 00:27:29,730 the difference of two squares, and we know that 353 00:27:29,730 --> 00:27:31,728 will give us a whole number. 354 00:27:32,890 --> 00:27:38,454 But we want to leave that this fraction has the same value, so 355 00:27:38,454 --> 00:27:43,590 we have to multiply it by the fraction 1 minus root 2. 356 00:27:44,800 --> 00:27:50,720 Over 1 minus root 2 because that fraction equals 1. 357 00:27:52,750 --> 00:27:57,282 So we evaluate what we've got now. Remember, this was the 358 00:27:57,282 --> 00:28:02,638 expansion of two squares and one of the two squares is 1 squared 359 00:28:02,638 --> 00:28:05,110 and the square root of 2. 360 00:28:06,200 --> 00:28:11,368 1 squared minus the square root of 2. 361 00:28:12,660 --> 00:28:16,734 That's what this product is when we 362 00:28:16,734 --> 00:28:20,808 evaluate that that equals 1 - 2. 363 00:28:22,010 --> 00:28:24,558 Which equals negative one. 364 00:28:27,340 --> 00:28:29,830 Now we simplify all of this. 365 00:28:30,330 --> 00:28:37,514 1 * 1 minus square root of 2 is 1 minus the square root of 2 366 00:28:37,514 --> 00:28:39,310 all over negative one. 367 00:28:40,590 --> 00:28:42,640 Now that doesn't look really 368 00:28:42,640 --> 00:28:48,128 neat. What we will do now is meeting it up. We divide through 369 00:28:48,128 --> 00:28:52,297 by the negative one and we divide through by the negative 370 00:28:52,297 --> 00:28:56,780 one. We get negative one into one which is negative one. 371 00:28:58,210 --> 00:29:00,500 Than negative one into the. 372 00:29:01,040 --> 00:29:06,116 Negative Route 2 will give us Plus Route 2. 373 00:29:07,100 --> 00:29:11,420 And we'll just rewrite it so that we've got the 374 00:29:11,420 --> 00:29:14,444 positive number. First, Route 2 subtract 1. 375 00:29:15,770 --> 00:29:20,016 So when we rationalize this fraction, one over 1 plus the 376 00:29:20,016 --> 00:29:25,806 square root of 2, you get the square root of 2 - 1 is not 377 00:29:25,806 --> 00:29:30,052 really neat. It looks great. You've got a fraction here, but 378 00:29:30,052 --> 00:29:31,982 here you've just got a 379 00:29:31,982 --> 00:29:34,620 subtraction. 1 380 00:29:34,620 --> 00:29:39,780 final example. More difficult than the previous 381 00:29:39,780 --> 00:29:44,652 ones, see how we can cope with this one over the square root of 382 00:29:44,652 --> 00:29:47,088 5 minus the square root of 3. 383 00:29:47,860 --> 00:29:49,960 We want to rationalize this 384 00:29:49,960 --> 00:29:53,739 fraction. Think of the difference of two squares. 385 00:29:54,500 --> 00:29:57,788 We've got ripped 5 minus Route 386 00:29:57,788 --> 00:30:05,404 3. If we multiply it by Route 5 Plus Route 3, we know 387 00:30:05,404 --> 00:30:09,580 that that expansion is the difference two squares. 388 00:30:10,500 --> 00:30:14,372 We don't want to change the value of the original fraction, 389 00:30:14,372 --> 00:30:18,596 so we have to multiply it by another fraction which equals 1 390 00:30:18,596 --> 00:30:24,228 and that will be square root of 5 + 3 over the square root of 5 391 00:30:24,228 --> 00:30:26,340 plus the square root of 3. 392 00:30:28,180 --> 00:30:30,160 We work guys. 393 00:30:30,780 --> 00:30:34,605 This expansion using the difference of two squares, the 394 00:30:34,605 --> 00:30:38,430 two squares involved are the square root of 5. 395 00:30:38,960 --> 00:30:45,908 Subtract the square root of 3. 396 00:30:48,450 --> 00:30:52,118 When we evaluate that that is 5 397 00:30:52,118 --> 00:30:56,010 - 3. Which equals 2. 398 00:30:58,100 --> 00:31:05,324 So when we rewrite all of this one times, the square root of 5 399 00:31:05,324 --> 00:31:10,484 plus the square root of 3 on top over 2. 400 00:31:11,530 --> 00:31:13,119 We tally it up a little bit. 401 00:31:13,680 --> 00:31:17,544 By dividing through by the two and so we get the square 402 00:31:17,544 --> 00:31:21,730 root of 5 over 2 plus the square root of 3 over 2. 403 00:31:23,680 --> 00:31:26,170 So that's it. That's how you 404 00:31:26,170 --> 00:31:29,870 rationalize. Search You have to. 405 00:31:30,370 --> 00:31:33,690 Make this error denominator into a whole number. 406 00:31:34,530 --> 00:31:37,110 By multiplying the fraction. 407 00:31:37,830 --> 00:31:40,290 By another fraction, which is one. 408 00:31:41,840 --> 00:31:44,890 And it's helpful to remember the difference of two squares. 409 00:31:45,630 --> 00:31:50,955 Now I want to want to do is finish off by giving you my top 410 00:31:50,955 --> 00:31:55,570 tents on serves. First of all recap what is assert assert is a 411 00:31:55,570 --> 00:31:57,345 fractional root of a whole 412 00:31:57,345 --> 00:32:02,719 number. Like a square root or cube root, the gives you an 413 00:32:02,719 --> 00:32:06,509 irrational number and remember an irrational number is a number 414 00:32:06,509 --> 00:32:10,678 that cannot be written as a whole number or a fraction. 415 00:32:11,330 --> 00:32:16,205 Remember there are common serves that we use a lot as route to 416 00:32:16,205 --> 00:32:18,830 Route 3, Route 5 and so on. 417 00:32:19,850 --> 00:32:24,790 And there's certain basic rules that we know that are 418 00:32:24,790 --> 00:32:28,742 useful when we are doing calculations involving Surds. 419 00:32:29,990 --> 00:32:33,446 One of them is if you have the square root of product. 420 00:32:34,560 --> 00:32:36,912 Then that can equal the square 421 00:32:36,912 --> 00:32:41,730 root. Of one number times by the square root of the other number. 422 00:32:42,770 --> 00:32:45,786 And similarly, the square root of a quotient. 423 00:32:46,930 --> 00:32:51,394 Is the square root of 1 number divided by the other square 424 00:32:51,394 --> 00:32:56,258 root? But remember the square root of addition is 425 00:32:56,258 --> 00:33:01,054 not the sum of the square roots that a common mistake, 426 00:33:01,054 --> 00:33:02,798 so don't make it. 427 00:33:04,280 --> 00:33:07,780 Another technique that we use 428 00:33:07,780 --> 00:33:10,695 insert calculations. Is 429 00:33:10,695 --> 00:33:17,160 rationalization? We don't like surge being the denominators of 430 00:33:17,160 --> 00:33:21,460 fractions. So what we have to do is rationalize the fraction. 431 00:33:22,280 --> 00:33:26,180 We multiply by a fraction that is equal to 1. 432 00:33:27,340 --> 00:33:29,804 And sometimes when we have to do that. 433 00:33:30,910 --> 00:33:35,508 The difference of two squares is useful. The expansion of the 434 00:33:35,508 --> 00:33:37,598 difference of two squares is. 435 00:33:38,170 --> 00:33:44,418 A squared minus B squared is a minus B Times A+B. 436 00:33:45,400 --> 00:33:50,197 Now, if you know all of those top tips, I think you'd be 437 00:33:50,197 --> 00:33:52,780 pretty good in doing calculations involving surds.