Exponents with negative bases
Let's see if we can apply what we know about negative numbers, and what we know about exponents to apply exponents to negative numbers. So let's first think about – Let's say we have -3. Let's first think about what it means to raise it to the 1st power. Well that literally means just taking a -3. And there's nothing left to multiply it with. So this is just going to be equal to -3. Now what happens if you were take a -3, and we were to raise it to the 2nd power? Well that's equivalent to taking 2 -3's, so a -3 and a -3, and then multiplying them together. What's that going to be? Well a negative times a negative is a positive. So that is going to be positive 9. Let me write this. It's going to be positive 9. Well, let's keep going. Let's see if there is some type of pattern here. Let's take -3 and raise it to the 3rd power. What is this going to be equal to? Well, we're going to take 3 -3's, [WRITING] – and we're going to multiply them together. So we're going to multiply them together. -3 × -3, we already figured out is positive 9. But positive 9 × -3, well that's that's -27. And so you might notice a pattern here. Whenever we raised raised a negative base to an exponent, if we raise it to an odd exponent, we are going to get a negative value. And that's because when you multiply negative numbers an even number of times, a negative number times a negative number is a positive. But then you have one more negative number to multiply the result by – which makes it negative. And if you take a negative base, and you raise it to an even power, that's because if you multiply a negative times a negative, you're going to get a positive. And so when you do it an even number of times, doing it a multiple-of-two number of times. So the negatives and the negatives all cancel out, I guess you could say. Or when you take the product of the two negatives, you keep getting positives. So this right over here is going to give you a positive value. So there's really nothing new about taking powers of negative numbers. It's really the same idea. And you just really have to remember that a negative times a negative is a positive. And a negative times a positive is a negative, which we already learned from multiplying negative numbers. Now there's one other thing that I want to clarify – because sometimes there might be ambiguity if someone writes this. Let's say someone writes that. And I encourage you to actually pause the video and think about with this right over here would evaluate to. And, if you given a go at that, think about whether this should mean something different then that. Well this one can be a little bit and big ambiguous and if people are strict about order of operations, you should really be thinking about the exponent before you multiply by this -1. You could this is implicitly saying -1 × 2^3. So many times, this will usually be interpreted as negative 2 to the third power, which is equal to -8, while this is going to be interpreted as -2 to the third power. Now that also is equal to -8. You might say well what's what's the big deal here? Well what if this was what if these were even exponents. So what if someone had give myself some more space here. What if someone had these to express its -4 or a -4 squared or -4 squared. This one clearly evaluates to 16 – positive 16. It's a negative 4 times a *4. This one could be interpreted as is. Especially if you look at order of operations, and you do your exponent first, this would be interpreted as -4 times 4, which would be -16. So it's really important to think about this properly. And if you want to write the number negative if you want the base to be negative 4, put parentheses around it and then write the exponent.