### Stony Brook MAT 123 Fall 2015 Rules of Exponents

 Start ok this module is about rules of exponents so what is X to power mean? it means you take X multiplied it by itself a times for example X to the fifth is x x x x x five times what if i want to find the x to the a times X to the 0:37 b remember X to the a means i take x x x multiplied it up a times and then this is going to be x to the b 1:01 the practical effect is just a long string of X is how many x's do i have a of them and have b of them i'm going to have x 8 plus B of them that tells me that X to the a times x to the B is next to the a plus B that's our first rule for example x to the third times x 4 x 3 plus 4 which is x to 1:30 the 7 notice it's not three times four so remember X cubed is X to the third the same as X cubed and x to the fourth is x 4 x so just roll it out that would be x x x x x x x, 7 times whish is x to the 7 okay second rule what about x to the a divided by x to the b. well now i have x times x 2:15 times x a time, divided by x times x times x b times so i'm dividing i'm going to cancel all 2:32 those X's and i'm just going to be left over so what's gonna be left over is a minus B x to the a over X to b so xto the a minus b so if i have x to the 8/ x to the 3 thats x to the 8-x to the 3 which is X to the fifth and we can convince ourselves that's true because x to the 8, 8 x's on top 3:08 to the three is three x's on the bottom those cancel and you're left with 5 x's now you may say what happened that more x's on the bottom then on the top well this is gonna come out a negative number so what does that mean to have a negative number well that would be 1 over so let's just do an example suppose i had X to the 3 over x to the 5 X to the 3:39 negative 2 so what does that look like y x times X times X on top x times X times X times X on the bottom 3 x's cancel left with 1 over s square so X to the minus 2 must be one over X to he poritive 2 this is another rule 4:06 that says x to the negative a is 1 over X to the positive a next rule, what if i have x to the positive a raised to the a thats x to the a times X to the a times X to the a x to the a B times so 4:43 I'm going to have X to the a/b time so remember this is X to the a+ a+ a+ a B times X to the a times B for example X square cube that's going to be best to 5:01 thats gonna be x to the 2 times 3 which is x to the 6 because that x squared cubed is x squared times x squared times x squared which is X to the two plus two plus two notices thats 3 of them which is X to the six ok let's go back to the division for a second what if the power of the top of the bottom is the same 5:32 what if have X to the a divided a to the a that would be X to the a minus a which ould be x to the 0 since the x's are the same that's just come out 1 so X to the 0 is always one so say i had x cubed on top and x cubed on the bottom, which is x to the 0 which is 1 been wondering y 0 power is one it means the power on top of the power on 6:05 the bottom are the same so let's make a list of the rules we've learned so far we learned that X to the a times X B is X to be a plus B and X to the a divided by x to the b is x to the a minus b, where x to the b is x to the a times b and x to the minus a 6:41 or x to the positive a and we learned that x to the 0 is positive 1 ok suppose I want to find I knew that X how do i want to show a root was like know that X cubed equals 7:11 a so I say to myself for what power x is x alone is what power a well i have to take something and multiplied it by itself three times a that gives me a thats each of these individually must be the cubed root of a because the cubed root of a times the 7:34 cube root of a times the cube root of a equals a so that means that this would be what power well the third power because if i take a to the 1/3 times a to the 1/3 times a to the one-third and you get one-third plus one-third plus one-third which is one which is a to the one so in general if i want to find X to the 1 over a thats a root 8:14 of x many people confuse the negative power with the root power as a reminder x to the negative 2 is one over x square X to the one-half is the square 8:33 root of x so don't confuse the fact that this is a fraction with this fraction