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# Exponents with negative bases

CCSS.Math:

## Video transcript

let's see if we can apply what we know about negative number what we know about negative numbers and we know about exponents to apply exponents to negative numbers so let's first think about let's just must say we have negative three let's first think about what it means to raise it to the first power well that literally means just taking a negative three just taking a negative three and there's nothing left to multiply it with so this is just going to be equal to negative three now what happens if we were to take a negative three and we were to raise it to the second power well that's equivalent to taking two negative 3s so negative three and a negative three and then multiplying them together and then multiplying them together and what's that going to be well a negative times a negative is a positive so that is going to be positive nine let me write this is going to be positive nine well let's keep going let's see if there's some type of a pattern here let's take negative three and raise it to the third power to the third power what is this going to be equal to well we're going to take three negative three one two and three now we're going to multiply them together so we're going to multiply them together negative three times negative three we already figured out is positive nine but positive nine times negative three well that's negative that's negative twenty-seven and so you might notice a pattern here whenever we raised whenever we raised a negative base so we raise a negative base to an exponent if we raise it to an odd exponent to an odd exponent we are going to get we are going to get a negative negative value and that's because when you multiply an even number of times the negatives a negative times a negative is a positive and then you have one more negative to multiply it by which makes it negative and if you take a negative base and you raise it to an even power and you raise it to an even power that's because you're always if you multiply a negative times a negative you're going to get a positive and then you're doing so when you do it an even number of times you're doing a multiple of two of times so the negatives and the negatives all cancel out I guess you could say or they when you take the product of the two negatives you keep getting positives so this is going to be this right over here is going to give you a positive value so there's really nothing new about multiplying or 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 let's say someone writes this let's say someone writes that I encourage you to actually pause your video and think about what this right over here would evaluate to and if you've given a go with that think about whether this should mean something different than that well this one can be a little bit ambiguous and if people are strict about order of operations you should really be thinking about the exponent before you multiply by this negative one you could this is implicitly saying negative 1 times 2 to the third power so many times this will usually be interpreted as negative 2 to the third power which is equal to negative 8 while this is going to be interpreted as negative 2 to the third power now that also is equal to negative 8 and 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 two expressions negative 4 or a negative 4 squared or negative 4 squared this one clearly evaluates to 16 positive 16 it's a negative 4 times a negative 4 this one could be interpreted as especially if you look at order of operations and you do your exponent first this would be interpreted as negative 4 times 4 which would be negative 16 so it's really important to think about this properly and if you want to write the number Nigg if you want the base to be negative or put parentheses around it and then write the exponent