Exponents with Negative Bases | Math with Mr. J

[Music] welcome to math with mr j [Music] in this video i'm going to cover exponents with negative bases remember an exponent tells us how many times we multiply the base by itself so let's jump into our examples here starting with number one where we have negative five squared we have a negative five that is within parentheses and then a negative 5 without parentheses there is a difference here so i want to show you how these differ the parentheses are very important so something to keep in mind let's start with the negative 5 that's within the parentheses now those parentheses indicate that the negative is included or is attached to our base so we have negative 5 times negative 5. let's write that out so negative 5 times negative 5. a negative times a negative equals a positive so this is going to give us a positive twenty-five let's compare that to the negative five without parentheses so here that negative is not included or attached it's not locked to the 5 so to speak so it stays in front this means the same thing as negative 1 times 5 squared or negative and then we have our 5 squared and the negative just stays in front so let's see what those different options look like so we can think of it as negative one times five squared or we can think of it as we have a negative in front it's not attached and then we have the 5 squared so let's start with the negative 1 times 5 squared well positive 5 times 5 5 squared that gives us 25 we have to do that 5 squared first due to the order of operations that comes before multiplication so 5 squared 5 times 5 equals a positive 25 then we bring down our negative one so we have negative one times 25 well a negative times a positive equals a negative so this is going to give us negative 25. that negative stays in front or we can think of it as the negative staying in front here and we have 5 squared we do that first and that negative again just stays in front so 5 squared is 25 and then the negative is not attached it stays in front so we get negative 25 a couple different ways of thinking about it there but the main point being the negative is not attached to that five it just stays in front so our solution or answer here is going to be negative no matter what now as far as with the parentheses if we have a negative times a negative we end up with a positive now there are some cases where we will end up with a negative no matter what whether we have parentheses or no parentheses and we're going to take a look at number two to see what that looks like so let's go on to number two where we have negative two cubed so let's start with the one with parentheses so the negative is attached to the two so this means negative two times negative two times negative two let's write that out so negative two times negative 2 times negative 2. negative 2 times negative 2 well a negative times a negative equals a positive so we get a positive four here and then we have one more negative two so a positive four times a negative two well a positive times a negative equals a negative so this is going to give us negative eight now for this one we're going to get the same exact answer without parentheses because remember that negative is going to stay in front so our solution or answer is going to automatically be negative let's see what that looks like essentially what we have here is negative 1 times 2 cubed a positive 2 cubed the negative is not attached to the 2 well a positive 2 cubed is 2 times 2 times 2 2 times 2 is 4 times 2 is a positive 8. so a positive 8 and then we still have our negative 1 times that positive 8 well a negative times a positive equals a negative so we end up with negative 8 there as well so in that case the parentheses do not impact our solution or answer because it's going to be negative either way another way to think of that real quick would be leaving the negative in front and then we have 2 cubed remember the negative is not attached to that two so we have two cubed which is eight and then that negative is in front so we end up with negative eight so a couple of different ways to think through that as far as leaving that negative sign in front and we automatically get that negative solution or answer there let's move on to number three where we have negative three to the power of four and the negative three is within parentheses so the negative is attached to that three it's included with the three so we have negative three times negative 3 times negative 3 times negative 3. so let's see what this equals well negative 3 times negative 3 a negative times a negative gives us a positive 9 then we have times negative 3 times negative 3 a positive 9 times a negative 3 well a positive times a negative equals a negative so this is going to give us negative 27 times a negative 3. so we end up with a negative times a negative which equals a positive so negative 27 times negative 3 equals a positive 81. a positive solution or answer there for number three and lastly let's move on to number four where we have negative 10 cubed and our negative 10 is within parentheses so this means negative 10 times negative 10 times negative 10. negative 10 times negative 10 a negative times a negative equals a positive so this is going to be a positive 100 and then we have times negative 10. we have a positive times a negative so we're going to get a negative 100 times negative 10 equals negative 1000 and that is our solution or answer there so there you have it there's how we evaluate exponents with negative bases i hope that helped thanks so much for watching until next time peace you