hey everybody welcome to my second video on exponents and in my first video I gave you a brief introduction to exponents and I went over some basic examples and in this video i want to go over negative exponents and also exponents on top of fractions so in my previous video I went over basic examples like 3 squared and that 2 exponent just means that we multiply 3 by itself 2 times 3 times 3 is equal to 9 so 3 squared is equal to 9 but what happens if we have a 3 with a negative 2 exponent well if you've never seen a negative exponent I think the natural logic would be if if 3 with a positive 2 exponent is is positive 9 then 3 with a negative 2 exponent would be negative 9 and if you use that logic then you are going to be completely wrong 100% of the time so how do we solve negative exponents well the first thing I'm going to do before I solve the negative exponent is x 1 multiplying by 1 does not change a thing a 3 with a negative 2 exponent is exactly the same as 1 times 3 with a negative 2 exponent and you don't have to multiply by 1 but I like to do it when I'm teaching because it makes it easier to see what's happening later on in this problem so now the question is how do we solve these negative exponents and what I like to do is make the negative exponent positive and there's only one thing you have to remember a negative exponent in the numerator is a positive exponent in the denominator so this 3 with a negative 2 exponent which is in the numerator right now can be re-written as a 3 with a positive exponent in the denominator and everything else stays the same the one which is in the numerator stays in the numerator so now we don't have anymore negative exponents we have 1 over three squared which is one over three x itself two times which is equal to 1 over 3 times 3 which is 9 1 over 9 so our three with a negative 2 exponent is equal to 1 over 9 all right let's go over another example let's say we have a negative exponent in the denominator let's say we have 1 over 3 with a negative 2 exponent and we're going to use the same idea as we did before we want to take this negative exponent and make it positive and we're going to use the same rule as we did before except this way this time we're going to use it the opposite way a negative exponent in the denominator is a positive exponent in the numerator so this 3 with a negative 2 exponent in the denominator is a 3 with a positive 2 exponent in the numerator and we can't forget about this one which is also in the numerator as well but multiplying by 1 is is kind of redundant it doesn't change the problem so this can just be simplified to 3 squared 3 squared is equal to 3 times 3 3 times 3 is equal to 9 so 1 over 3 to the negative 2 exponent is equal to positive 9 all right let's go over another example this time with some negative numbers and before I do that I'm going to erase everything so let's say we had 6 multiplied by negative 3 with a negative 2 exponent and remember a negative exponent in the numerator is a positive exponent in the denominator so this negative 3 with a negative 2 exponent in the numerator can be written as a negative 3 with a positive 2 exponent in the denominator and everything else stays the same the 6 which is in the numerator stays in the numerator so now this fraction can be simplified even further negative 3 with a 2 exponent is just negative 3 multiplied by itself two times and in our numerator we have a 6 a negative 3 times a negative 3 is going to give us a positive 9 and in our numerator we have a 6 and 6 over 9 can be reduced to 2 over 3 so 6 multiplied by negative 3 with a negative 2 exponent can't be simplified and our answer is 2/3 all right let's go over one more example this time with a fraction and I think after this you should have a really good idea about negative exponents so so let's say we have 3 over 4 with a negative 2 exponent and once again I'm going to multiply by 1/3 over 4 with a negative 2 exponent is exactly the same as 1 times 3 over 4 with a negative 2 exponent so now let's get rid of this negative exponent a negative exponent in the numerator is a positive exponent in the denominator so right now this is 3 over 4 with a negative 2 exponent is in the numerator so we could rewrite this as a a 3 over 4 with a positive 2 exponent in the denominator and everything else stays the same the one which is in the numerator stays in the numerator so now in the denominator we have 3 over 4 squared so that is just 3 over 4 multiplied by itself 2 times 3 over 4 times 3 over 4 and everything else stays the same we still have our 1 in the numerator and if we simplify our denominator we have 3 times 3 which is equal to 9 and we have 4 times 4 which is equal to 16 and in our numerator we still have our 1 so now we're dividing one by a fraction whenever you're dividing by a fraction that is same as multiplying by the reciprocal so 1/9 over 16 is the same as 1 multiplied by 16 over 9 dividing by a fraction is the same as multiplying by the reciprocal and this could be simplified even further 16 over 9 times 1 is just 16 over 9 so this is our final answer and I hope this gave you a better idea on negative exponents in fractions in my next video I'm going to talk about the properties of exponents exponent properties so stay tuned for that I really hope you're enjoying these and I will see you in my next one