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
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
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
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
times x a time, divided by x times x times x b times so i'm dividing i'm going to cancel all
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
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
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
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
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
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
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
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
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
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
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
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
root of x so don't confuse the fact that this is a fraction with this fraction