1 00:00:00,372 --> 00:00:03,906 What I want to do in this video is simplify this expression 2 00:00:03,906 --> 00:00:06,845 3 times the principal root of 500 times "X to the third" 3 00:00:06,845 --> 00:00:11,190 and take it into consideration some of the comments that we got out on YouTube that actually 4 00:00:11,190 --> 00:00:15,282 uh, gave some interesting perspective on how you can simplify this. 5 00:00:15,282 --> 00:00:21,146 So this is a quick review of what we did in the last video, we said that this is the same thing as 3 times 6 00:00:21,146 --> 00:00:25,797 the principal root of 500, and I'm gonna do it a little bit different than I did in the last video, just, just 7 00:00:25,797 --> 00:00:27,491 so to make it interesting. 8 00:00:27,491 --> 00:00:33,210 This is 3 times the principal root of 500 times the principal root of "X to the third." 9 00:00:33,210 --> 00:00:42,181 And, 500, we can rewrite, because 500 is not a perfect square, we can rewrite 500 as 100 times 10 00:00:42,181 --> 00:00:47,436 5. Or even better, we can rewrite that as 10 squared times 5... 10 squared is the same thing as 100. 11 00:00:47,436 --> 00:00:55,779 So we can rewrite this first part over here 3 times the principal root of 10 squared times 5, 12 00:00:55,779 --> 00:01:03,033 times, times the principal root of x squared times x. 13 00:01:03,033 --> 00:01:04,774 That's the same thing as x to the third. 14 00:01:04,774 --> 00:01:08,327 Now the one thing I'm gonna do here... actually I won't talk about it just yet, of how we are gonna 15 00:01:08,327 --> 00:01:10,051 do it differently in the last video. 16 00:01:10,051 --> 00:01:12,596 This radical, right here, can be rewritten as... 17 00:01:12,596 --> 00:01:17,477 So this is gonna be three times the square root, or the principal root, I should say, of 10 squared times... 18 00:01:17,477 --> 00:01:20,361 the square root of 5. 19 00:01:20,361 --> 00:01:24,513 If we take the square root of the product of two things, it's the same thing as taking the square root of each of... 20 00:01:24,513 --> 00:01:26,644 them, and then taking the product. 21 00:01:26,644 --> 00:01:32,412 And, so in this over here is going to be times the square root of where the principal root of x squared times... 22 00:01:32,412 --> 00:01:35,992 the principal root of "X." 23 00:01:35,992 --> 00:01:43,180 And, the principal root of 10 squared is 10, and then what I said in the last video was the principal root... 24 00:01:43,180 --> 00:01:46,716 of "X" squared is going to be the absolute value of "X." 25 00:01:46,716 --> 00:01:50,814 Just in case, just in case, if "X" itself is a negative number. 26 00:01:50,814 --> 00:01:56,777 So, then if you simplify all this, you get 3 times 10, which is 30 times, and I'm just gonna switch... 27 00:01:56,777 --> 00:02:01,883 the order here, times the absolute value of X, and then you have this square root of 5, 28 00:02:01,883 --> 00:02:05,530 or the principal root of 5, times the principal root of X. 29 00:02:05,530 --> 00:02:09,898 And this is just going to be equal to the principal root of 5x. 30 00:02:09,898 --> 00:02:13,511 Taking the square root of something, and multiplying that times the square root of something else... 31 00:02:13,511 --> 00:02:16,650 is the same thing as just to taking the square root of 5x. 32 00:02:16,650 --> 00:02:22,148 So, all of this simplified down to 30 times the absolute value of X, times the principal root of... 33 00:02:22,148 --> 00:02:23,045 5x. 34 00:02:23,045 --> 00:02:24,459 And this is what we got in the last video. 35 00:02:24,459 --> 00:02:29,463 And the interesting thing here is, if we assume we're only dealing with real numbers, the domain... 36 00:02:29,463 --> 00:02:36,407 of X, right over here, the X's that will make this expression define in the real numbers... 37 00:02:36,407 --> 00:02:41,830 then, X has to be, greater than or equal to 0. 38 00:02:41,830 --> 00:02:44,783 So, let me, so maybe I could write it this way. 39 00:02:44,783 --> 00:02:58,643 The domain, here, is that X is any real number, greater than or equal to 0. 40 00:02:58,643 --> 00:03:03,863 The reason why I said that is that you put a negative number in here, and you cube it, you're... 41 00:03:03,863 --> 00:03:05,464 gonna get another negative number. 42 00:03:05,464 --> 00:03:13,229 And, then it doesn't make it, at least in the real numbers, you won't get an actual value. 43 00:03:13,229 --> 00:03:15,367 You'll get a square root of a negative number. 44 00:03:15,367 --> 00:03:19,745 So, if you make this, if you assume this right here, we're dealing with the real numbers... 45 00:03:19,745 --> 00:03:24,808 we're not dealing with any complex numbers, when you hope their not complex numbers, then... 46 00:03:24,808 --> 00:03:28,416 you can have a, you can expand the domain more broadly. 47 00:03:28,416 --> 00:03:32,461 If you're dealing with real numbers, you can say that X is going to be greater than or equal to... 48 00:03:32,461 --> 00:03:33,429 0. 49 00:03:33,429 --> 00:03:36,645 And, then, the absolute value of X is just going to be X, 'cause it's not going to be... 50 00:03:36,645 --> 00:03:38,041 a negative number. 51 00:03:38,041 --> 00:03:44,472 And, if, so assuming that the domain of X is, er, if this expression is going to be invaluable, or it's... 52 00:03:44,472 --> 00:03:50,280 it's going to have a positive number, that this can be written as 30X times the square root of... 53 00:03:50,280 --> 00:03:51,551 ... of 5X. 54 00:03:51,551 --> 00:03:57,050 If you had the situation, where we're dealing with complex numbers, then you would... 55 00:03:57,050 --> 00:04:01,582 So numbers that were, and if you don't know what a complex number is, or an imaginary number, don't... 56 00:04:01,582 --> 00:04:03,167 worry too much about it. 57 00:04:03,167 --> 00:04:07,433 But, if you were dealing with those, then you would have to keep the absolute value of X... 58 00:04:07,433 --> 00:04:11,433 there, because then this would be defined for numbers that are less than 0.