WEBVTT 00:00:00.405 --> 00:00:01.456 Let's see if we can apply 00:00:01.456 --> 00:00:04.532 what we know about negative numbers, 00:00:04.532 --> 00:00:06.031 and what we know about exponents 00:00:06.031 --> 00:00:08.879 to apply exponents to negative numbers. 00:00:08.879 --> 00:00:10.197 So let's first think about – 00:00:10.197 --> 00:00:13.723 Let's say we have -3. 00:00:13.723 --> 00:00:15.123 Let's first think about what it means 00:00:15.123 --> 00:00:18.057 to raise it to the 1st power. 00:00:18.057 --> 00:00:22.085 Well that literally means just taking a -3. 00:00:22.085 --> 00:00:23.994 And there's nothing left to multiply it with. 00:00:23.994 --> 00:00:27.642 So this is just going to be equal to -3. 00:00:27.642 --> 00:00:30.069 Now what happens if you were take a -3, 00:00:30.069 --> 00:00:34.144 and we were to raise it to the 2nd power? 00:00:34.144 --> 00:00:37.660 Well that's equivalent to taking 2 -3's, 00:00:37.660 --> 00:00:44.661 so a -3 and a -3, and then multiplying them together. 00:00:44.661 --> 00:00:46.457 What's that going to be? 00:00:46.457 --> 00:00:48.930 Well a negative times a negative is a positive. 00:00:48.930 --> 00:00:53.670 So that is going to be positive 9. 00:00:53.670 --> 00:00:55.918 Let me write this. It's going to be positive 9. 00:00:55.918 --> 00:00:56.798 Well, let's keep going. 00:00:56.798 --> 00:00:59.054 Let's see if there is some type of pattern here. 00:00:59.054 --> 00:01:04.798 Let's take -3 and raise it to the 3rd power. 00:01:04.798 --> 00:01:06.921 What is this going to be equal to? 00:01:06.921 --> 00:01:12.659 Well, we're going to take 3 -3's, [WRITING] – 00:01:12.659 --> 00:01:14.629 and we're going to multiply them together. 00:01:14.629 --> 00:01:16.582 So we're going to multiply them together. 00:01:16.582 --> 00:01:20.195 -3 × -3, we already figured out is positive 9. 00:01:20.195 --> 00:01:26.806 But positive 9 × -3, well that's that's -27. 00:01:26.806 --> 00:01:28.720 And so you might notice a pattern here. 00:01:28.720 --> 00:01:33.057 Whenever we raised raised a negative base 00:01:33.057 --> 00:01:41.188 to an exponent, if we raise it to an odd exponent, 00:01:41.188 --> 00:01:47.910 we are going to get a negative value. 00:01:47.910 --> 00:01:49.402 And that's because when you multiply 00:01:49.402 --> 00:01:50.853 negative numbers an even number of times, 00:01:50.853 --> 00:01:52.122 a negative number times 00:01:52.122 --> 00:01:53.006 a negative number is a positive. 00:01:53.006 --> 00:01:54.521 But then you have one more negative number 00:01:54.521 --> 00:01:56.696 to multiply the result by – which makes it negative. 00:01:56.696 --> 00:01:59.594 And if you take a negative base, 00:01:59.594 --> 00:02:04.471 and you raise it to an even power, 00:02:04.471 --> 00:02:07.327 that's because if you multiply a negative 00:02:07.327 --> 00:02:09.726 times a negative, you're going to get a positive. 00:02:09.726 --> 00:02:12.055 And so when you do it an even number of times, 00:02:12.055 --> 00:02:14.133 doing it a multiple-of-two number of times. 00:02:14.133 --> 00:02:16.736 So the negatives and the negatives all cancel out, 00:02:16.736 --> 00:02:17.529 I guess you could say. 00:02:17.529 --> 00:02:19.735 Or when you take the product of the two negatives, 00:02:19.735 --> 00:02:21.267 you keep getting positives. 00:02:21.267 --> 00:02:23.867 So this right over here is 00:02:23.867 --> 00:02:27.660 going to give you a positive value. 00:02:27.660 --> 00:02:29.497 So there's really nothing new about 00:02:29.497 --> 00:02:32.489 taking powers of negative numbers. 00:02:32.489 --> 00:02:33.659 It's really the same idea. 00:02:33.659 --> 00:02:35.409 And you just really have to remember that 00:02:35.409 --> 00:02:38.136 a negative times a negative is a positive. 00:02:38.136 --> 00:02:41.663 And a negative times a positive is a negative, 00:02:41.663 --> 00:02:43.297 which we already learned from 00:02:43.297 --> 00:02:45.545 multiplying negative numbers. 00:02:45.545 --> 00:02:48.072 Now there's one other thing that I want to clarify – 00:02:48.072 --> 00:02:50.549 because sometimes there might be ambiguity 00:02:50.549 --> 00:02:54.227 if someone writes this. 00:02:54.227 --> 00:02:55.895 Let's say someone writes that. 00:02:55.895 --> 00:02:58.328 And I encourage you to actually pause the video 00:02:58.328 --> 00:03:01.472 and think about with this right over here 00:03:01.472 --> 00:03:03.332 would evaluate to. 00:03:03.332 --> 00:03:05.963 And, if you given a go at that, think about whether 00:03:05.963 --> 00:03:11.793 this should mean something different then that. 00:03:11.793 --> 00:03:14.664 Well this one can be a little bit and big ambiguous 00:03:14.664 --> 00:03:17.031 and if people are strict about order of operations, NOTE Paragraph 00:03:17.031 --> 00:03:19.469 you should really be thinking about the exponent 00:03:19.485 --> 00:03:21.786 before you multiply by this -1. 00:03:21.786 --> 00:03:25.895 You could this is implicitly saying -1 × 2^3. 00:03:25.895 --> 00:03:29.003 So many times, this will usually be interpreted 00:03:29.003 --> 00:03:32.060 as negative 2 to the third power, 00:03:32.060 --> 00:03:34.501 which is equal to -8, 00:03:34.501 --> 00:03:36.382 while this is going to be interpreted 00:03:36.382 --> 00:03:38.388 as -2 to the third power. 00:03:38.388 --> 00:03:40.525 Now that also is equal to -8. 00:03:40.525 --> 00:03:43.461 You might say well what's what's the big deal here? 00:03:43.461 --> 00:03:46.929 Well what if this was what if these were even exponents. 00:03:46.929 --> 00:03:51.486 So what if someone had give myself some more space here. 00:03:51.486 --> 00:04:00.614 What if someone had these to express its -4 or a -4 squared or -4 squared. 00:04:00.614 --> 00:04:04.558 This one clearly evaluates to 16 – positive 16. 00:04:04.558 --> 00:04:07.189 It's a negative 4 times a *4. 00:04:07.189 --> 00:04:09.073 This one could be interpreted as is. 00:04:09.073 --> 00:04:11.003 Especially if you look at order of operations, 00:04:11.003 --> 00:04:12.564 and you do your exponent first, 00:04:12.564 --> 00:04:15.766 this would be interpreted as -4 times 4, 00:04:15.766 --> 00:04:18.077 which would be -16. 00:04:18.077 --> 00:04:20.964 So it's really important to think about this properly. 00:04:20.964 --> 00:04:23.438 And if you want to write the number negative 00:04:23.438 --> 00:04:25.474 if you want the base to be negative 4, 00:04:25.474 --> 00:04:29.000 put parentheses around it and then write the exponent.