1 00:00:00,740 --> 00:00:04,160 Write the prime factorization of 75. 2 00:00:04,160 --> 00:00:07,390 Write your answer using exponential notation. 3 00:00:07,390 --> 00:00:08,970 So we have a couple of interesting things here. 4 00:00:08,970 --> 00:00:12,410 Prime factorization, and they say exponential notation. 5 00:00:12,410 --> 00:00:15,460 We'll worry about the exponential notation later. 6 00:00:15,460 --> 00:00:18,560 So the first thing we have to worry about is what is even a 7 00:00:18,560 --> 00:00:19,380 prime number? 8 00:00:19,380 --> 00:00:22,240 And just as a refresher, a prime number is a number 9 00:00:22,240 --> 00:00:26,130 that's only divisible by itself and one, so examples of 10 00:00:26,130 --> 00:00:28,880 prime numbers-- let me write some numbers down. 11 00:00:28,880 --> 00:00:34,753 Prime, not prime. Not prime. 12 00:00:34,760 --> 00:00:36,840 So 2 is a prime number. 13 00:00:36,840 --> 00:00:39,850 It's only divisible by 1 and 2. 14 00:00:39,850 --> 00:00:42,490 3 is another prime number. 15 00:00:42,490 --> 00:00:46,790 Now, 4 is not prime, because this is 16 00:00:46,790 --> 00:00:49,790 divisible by 1, 2 and 4. 17 00:00:49,790 --> 00:00:50,580 We could keep going. 18 00:00:50,580 --> 00:00:56,220 5, well, 5 is only divisible by 1 and 5, so 5 is prime. 19 00:00:56,220 --> 00:00:59,920 6 is not prime, because it's divisible by 2 and 3. 20 00:00:59,920 --> 00:01:01,590 I think you get the general idea. 21 00:01:01,590 --> 00:01:04,160 You move to 7, 7 is prime. 22 00:01:04,160 --> 00:01:06,470 It's only divisible by 1 and 7. 23 00:01:06,470 --> 00:01:08,220 8 is not prime. 24 00:01:08,220 --> 00:01:11,440 9 you might be tempted to say is prime, but remember, it's 25 00:01:11,440 --> 00:01:15,420 also divisible by 3, so 9 is not prime. 26 00:01:15,420 --> 00:01:18,970 Prime is not the same thing as odd numbers. 27 00:01:18,970 --> 00:01:21,400 Then if you move to 10, 10 is also not prime, 28 00:01:21,400 --> 00:01:23,560 divisible by 2 and 5. 29 00:01:23,560 --> 00:01:27,220 11, it's only divisible by 1 and 11, so 11 is 30 00:01:27,220 --> 00:01:28,240 then a prime number. 31 00:01:28,240 --> 00:01:29,780 And we could keep going on like this. 32 00:01:29,780 --> 00:01:31,570 People have written computer programs looking for the 33 00:01:31,570 --> 00:01:33,260 highest prime and all of that. 34 00:01:33,260 --> 00:01:35,220 So now that we know what a prime is, a prime 35 00:01:35,220 --> 00:01:39,240 factorization is breaking up a number, like 75, into a 36 00:01:39,240 --> 00:01:41,620 product of prime numbers. 37 00:01:41,620 --> 00:01:43,180 So let's try to do that. 38 00:01:43,180 --> 00:01:45,530 So we're going to start with 75, and I'm going to do it 39 00:01:45,530 --> 00:01:49,080 using what we call a factorization tree. 40 00:01:49,080 --> 00:01:51,750 So we first try to find just the smallest prime number that 41 00:01:51,750 --> 00:01:53,890 will go into 75. 42 00:01:53,890 --> 00:01:55,430 Now, the smallest prime number is 2. 43 00:01:55,430 --> 00:01:57,390 Does 2 go into 75? 44 00:01:57,390 --> 00:02:00,705 Well, 75 is an odd number, or the number in the ones place, 45 00:02:00,705 --> 00:02:02,280 this 5, is an odd number. 46 00:02:02,280 --> 00:02:06,580 5 is not divisible by 2, so 2 will not go into 75. 47 00:02:06,580 --> 00:02:08,090 So then we could try 3. 48 00:02:08,090 --> 00:02:09,639 Does 3 go into 75? 49 00:02:09,639 --> 00:02:12,440 Well, 7 plus 5 is 12. 50 00:02:12,440 --> 00:02:15,480 12 is divisible by 3, so 3 will go into it. 51 00:02:15,480 --> 00:02:20,440 So 75 is 3 times something else. 52 00:02:20,440 --> 00:02:22,990 And if you've ever dealt with change, you know that if you 53 00:02:22,990 --> 00:02:25,890 have three quarters, you have 75 cents, or if you have 3 54 00:02:25,890 --> 00:02:28,930 times 25, you have 75. 55 00:02:28,930 --> 00:02:31,560 So this is 3 times 25. 56 00:02:31,560 --> 00:02:33,720 And you can multiply this out if you don't believe me. 57 00:02:33,720 --> 00:02:35,960 Multiple out 3 times 25. 58 00:02:35,960 --> 00:02:40,470 Now, is 25 divisible by-- you can give up on 2. 59 00:02:40,470 --> 00:02:44,910 If 75 wasn't divisible by 2, 25's not going to be divisible 60 00:02:44,910 --> 00:02:46,000 by 2 either. 61 00:02:46,000 --> 00:02:48,730 But maybe 25 is divisible by 3 again. 62 00:02:48,730 --> 00:02:52,290 So if you take the digits 2 plus 5, you get 7. 63 00:02:52,290 --> 00:02:57,700 7 is not divisible by 3, so 25 will not be divisible by 3. 64 00:02:57,700 --> 00:02:59,480 So we keep moving up: 5. 65 00:02:59,480 --> 00:03:01,430 Is 25 divisible by 5? 66 00:03:01,430 --> 00:03:01,980 Well, sure. 67 00:03:01,980 --> 00:03:03,590 It's 5 times 5. 68 00:03:03,590 --> 00:03:08,330 So 25 is 5 times 5. 69 00:03:08,330 --> 00:03:11,730 And we're done with our prime factorization because now we 70 00:03:11,730 --> 00:03:13,390 have all prime numbers here. 71 00:03:13,390 --> 00:03:18,270 So we can write that 75 is 3 times 5 times 5. 72 00:03:18,270 --> 00:03:25,640 So 75 is equal to 3 times 5 times 5. 73 00:03:25,640 --> 00:03:27,350 We can say it's 3 times 25. 74 00:03:27,350 --> 00:03:29,400 25 is 5 times 5. 75 00:03:29,400 --> 00:03:33,370 3 times 25, 25 is 5 times 5. 76 00:03:33,370 --> 00:03:36,460 So this is a prime factorization, but they want 77 00:03:36,460 --> 00:03:41,690 us to write our answer using exponential notation. 78 00:03:41,690 --> 00:03:44,560 So that just means, if we have repeated primes, we can write 79 00:03:44,560 --> 00:03:45,920 those as an exponent. 80 00:03:45,920 --> 00:03:48,480 So what is 5 times 5? 81 00:03:48,480 --> 00:03:52,380 5 times 5 is 5 multiplied by itself two times. 82 00:03:52,380 --> 00:03:56,310 This is the same thing as 5 to the second power. 83 00:03:56,310 --> 00:03:58,380 So if we want to write our answer using exponential 84 00:03:58,380 --> 00:04:03,420 notation, we could say this is equal to 3 times 5 to the 85 00:04:03,420 --> 00:04:08,110 second power, which is the same thing as 5 times 5.