1 00:00:00,405 --> 00:00:01,456 Let's see if we can apply 2 00:00:01,456 --> 00:00:04,532 what we know about negative numbers, 3 00:00:04,532 --> 00:00:06,031 and what we know about exponents 4 00:00:06,031 --> 00:00:08,879 to apply exponents to negative numbers. 5 00:00:08,879 --> 00:00:10,197 So let's first think about – 6 00:00:10,197 --> 00:00:13,723 Let's say we have -3. 7 00:00:13,723 --> 00:00:15,123 Let's first think about what it means 8 00:00:15,123 --> 00:00:18,057 to raise it to the 1st power. 9 00:00:18,057 --> 00:00:22,085 Well that literally means just taking a -3. 10 00:00:22,085 --> 00:00:23,994 And there's nothing left to multiply it with. 11 00:00:23,994 --> 00:00:27,642 So this is just going to be equal to -3. 12 00:00:27,642 --> 00:00:30,069 Now what happens if you were take a -3, 13 00:00:30,069 --> 00:00:34,144 and we were to raise it to the 2nd power? 14 00:00:34,144 --> 00:00:37,660 Well that's equivalent to taking 2 -3's, 15 00:00:37,660 --> 00:00:44,661 so a -3 and a -3, and then multiplying them together. 16 00:00:44,661 --> 00:00:46,457 What's that going to be? 17 00:00:46,457 --> 00:00:48,930 Well a negative times a negative is a positive. 18 00:00:48,930 --> 00:00:53,670 So that is going to be positive 9. 19 00:00:53,670 --> 00:00:55,918 Let me write this. It's going to be positive 9. 20 00:00:55,918 --> 00:00:56,798 Well, let's keep going. 21 00:00:56,798 --> 00:00:59,054 Let's see if there is some type of pattern here. 22 00:00:59,054 --> 00:01:04,798 Let's take -3 and raise it to the 3rd power. 23 00:01:04,798 --> 00:01:06,921 What is this going to be equal to? 24 00:01:06,921 --> 00:01:12,659 Well, we're going to take 3 -3's, [WRITING] – 25 00:01:12,659 --> 00:01:14,629 and we're going to multiply them together. 26 00:01:14,629 --> 00:01:16,582 So we're going to multiply them together. 27 00:01:16,582 --> 00:01:20,195 -3 × -3, we already figured out is positive 9. 28 00:01:20,195 --> 00:01:26,806 But positive 9 × -3, well that's that's -27. 29 00:01:26,806 --> 00:01:28,720 And so you might notice a pattern here. 30 00:01:28,720 --> 00:01:33,057 Whenever we raised raised a negative base 31 00:01:33,057 --> 00:01:41,188 to an exponent, if we raise it to an odd exponent, 32 00:01:41,188 --> 00:01:47,910 we are going to get a negative value. 33 00:01:47,910 --> 00:01:49,402 And that's because when you multiply 34 00:01:49,402 --> 00:01:50,853 negative numbers an even number of times, 35 00:01:50,853 --> 00:01:52,122 a negative number times 36 00:01:52,122 --> 00:01:53,006 a negative number is a positive. 37 00:01:53,006 --> 00:01:54,521 But then you have one more negative number 38 00:01:54,521 --> 00:01:56,696 to multiply the result by – which makes it negative. 39 00:01:56,696 --> 00:01:59,594 And if you take a negative base, 40 00:01:59,594 --> 00:02:04,471 and you raise it to an even power, 41 00:02:04,471 --> 00:02:07,327 that's because if you multiply a negative 42 00:02:07,327 --> 00:02:09,726 times a negative, you're going to get a positive. 43 00:02:09,726 --> 00:02:12,055 And so when you do it an even number of times, 44 00:02:12,055 --> 00:02:14,133 doing it a multiple-of-two number of times. 45 00:02:14,133 --> 00:02:16,736 So the negatives and the negatives all cancel out, 46 00:02:16,736 --> 00:02:17,529 I guess you could say. 47 00:02:17,529 --> 00:02:19,735 Or when you take the product of the two negatives, 48 00:02:19,735 --> 00:02:21,267 you keep getting positives. 49 00:02:21,267 --> 00:02:23,867 So this right over here is 50 00:02:23,867 --> 00:02:27,660 going to give you a positive value. 51 00:02:27,660 --> 00:02:29,497 So there's really nothing new about 52 00:02:29,497 --> 00:02:32,489 taking powers of negative numbers. 53 00:02:32,489 --> 00:02:33,659 It's really the same idea. 54 00:02:33,659 --> 00:02:35,409 And you just really have to remember that 55 00:02:35,409 --> 00:02:38,136 a negative times a negative is a positive. 56 00:02:38,136 --> 00:02:41,663 And a negative times a positive is a negative, 57 00:02:41,663 --> 00:02:43,297 which we already learned from 58 00:02:43,297 --> 00:02:45,545 multiplying negative numbers. 59 00:02:45,545 --> 00:02:48,072 Now there's one other thing that I want to clarify – 60 00:02:48,072 --> 00:02:50,549 because sometimes there might be ambiguity 61 00:02:50,549 --> 00:02:54,227 if someone writes this. 62 00:02:54,227 --> 00:02:55,895 Let's say someone writes that. 63 00:02:55,895 --> 00:02:58,328 And I encourage you to actually pause the video 64 00:02:58,328 --> 00:03:01,472 and think about with this right over here 65 00:03:01,472 --> 00:03:03,332 would evaluate to. 66 00:03:03,332 --> 00:03:05,963 And, if you given a go at that, think about whether 67 00:03:05,963 --> 00:03:11,793 this should mean something different then that. 68 00:03:11,793 --> 00:03:14,664 Well this one can be a little bit and big ambiguous 69 00:03:14,664 --> 00:03:17,031 and if people are strict about order of operations, 70 00:03:17,031 --> 00:03:19,469 you should really be thinking about the exponent 71 00:03:19,485 --> 00:03:21,786 before you multiply by this -1. 72 00:03:21,786 --> 00:03:25,895 You could this is implicitly saying -1 × 2^3. 73 00:03:25,895 --> 00:03:29,003 So many times, this will usually be interpreted 74 00:03:29,003 --> 00:03:32,060 as negative 2 to the third power, 75 00:03:32,060 --> 00:03:34,501 which is equal to -8, 76 00:03:34,501 --> 00:03:36,382 while this is going to be interpreted 77 00:03:36,382 --> 00:03:38,388 as -2 to the third power. 78 00:03:38,388 --> 00:03:40,525 Now that also is equal to -8. 79 00:03:40,525 --> 00:03:43,461 You might say well what's what's the big deal here? 80 00:03:43,461 --> 00:03:46,929 Well what if this was what if these were even exponents. 81 00:03:46,929 --> 00:03:51,486 So what if someone had give myself some more space here. 82 00:03:51,486 --> 00:04:00,614 What if someone had these to express its -4 or a -4 squared or -4 squared. 83 00:04:00,614 --> 00:04:04,558 This one clearly evaluates to 16 – positive 16. 84 00:04:04,558 --> 00:04:07,189 It's a negative 4 times a *4. 85 00:04:07,189 --> 00:04:09,073 This one could be interpreted as is. 86 00:04:09,073 --> 00:04:11,003 Especially if you look at order of operations, 87 00:04:11,003 --> 00:04:12,564 and you do your exponent first, 88 00:04:12,564 --> 00:04:15,766 this would be interpreted as -4 times 4, 89 00:04:15,766 --> 00:04:18,077 which would be -16. 90 00:04:18,077 --> 00:04:20,964 So it's really important to think about this properly. 91 00:04:20,964 --> 00:04:23,438 And if you want to write the number negative 92 00:04:23,438 --> 00:04:25,474 if you want the base to be negative 4, 93 00:04:25,474 --> 00:04:29,000 put parentheses around it and then write the exponent.