BAM Mr. Tarrou well we got done talking about the multiplication property and
the power property of exponents now let's move on to the division property
now if you multiply like bases and when you multiply like bases if you add the
exponents guess what you do with division the inverse of multiplication
well it's the inverse of the inverse math operation of addition that's
subtraction so first example here we have X to the fifth over x squared now
this property says that when you divide like bases you're supposed to subtract
the exponents so you know this answer is supposedly X to the fifth
my excuse me X to the five minus two you subtract down so five minus two power
and that is equal to X to the third now if you don't believe that then you can
take this and let's just expand it X to the fifth power something in the fifth
power means you're multiplying by that base five times so there's one two three
four five X's in the denominator we have x squared so we have two x's well also
remember that anything divided by itself is equal to one so X divided by X is one
X divided by X is again one and x times X times X the last time I checked was
equal to X to the third now that's not a proof I'm just showing you that with
this expansion if you just understand what exponents do just by expanding
those exponents X to the fifth and x squared
you're going to get to the same answer so yes indeedy when you divide like
bases you subtract those exponents okay also as we go through this video and I
do have quite a few examples a couple of things one I'm going to make sure that
all of our answers have positive exponents because I'm going to deal with
negative exponents and powers of zero in the next video and explain to you how
the division property will allow you to see why is it anything that zero power
is equal to one this is an example also I'm not going to write a condition every
time I put up a new example so we're just going to assume that all these
variables are going to be set up or have restrictions of their in their domain so
that will never divide by 0 so like for this problem X can be equal to 0 or Y
can't be equal to 0 because the whole problem be undefined it'd be pointless
trying to simplify the fraction the rational expression so here we have X to
the seventh over X to the fourth to our just say this way X to the seventh times
y squared over X to the fourth times y to the first don't need to write that
one there but if you're just learning this you might need it now how do you do
this if you really go with this you know if you're if you're catching on this
really quick before long you will be able to do this in your head however I'm
just teaching it someone to show all the work we have a variable of X so I'm
going to I'm going to pair up all the variables that are in the numerator and
denominator or just a memory or a lung and sort of give them their own little a
little piece here so we're going to take the X and write that as X to the 7 minus
4 power times because along the top of this fraction in the bottom for that
matter but along the top of this fraction all of these factors are being
multiplied together you you can't do any of the all these examples were to do
today are going to have one term on the numerator and one term in the
denominator if you start having plus and minus in here that's quite a bit of a
different story how to simplify those fractions so we have X to the 7 minus 4
because we're dividing like bases we're dividing the Y's there's a common base
of Y in the numerator and denominator so it's going to be y to the 2 minus 1
power and X to the 7 minus 4 is X to the 3rd and 2 minus 1 is equal to 1 so
cubed times fly to the first really good you don't really have to write that here
we have a fraction and the whole fraction is being raised to the second
power well power to power property anytime you have those stacked exponents
you are going to need to apply the power property and again remember there's no
pluses and minus in the numerator or denominator it's one term over one term
so because of that and because of the fact that we have these stacked
exponents I'm going to go ahead and put gives us three a exponent one so we can
see it and we're going to kind of like to do the distributive property we're
doing the the power property every single one of the exponents that are in
this fraction again where there's a monomial on top and a monomial in the
denominator each one of those exponents are going to get that power of two so
I'm going to do the power to power to each one of those each one of those
exponents okay so we have three one times two is two so we have three
squared X cubed now that three times two is equal to six we're going X to the
sixth over negative not a crab neck negative sign up in a set of parentheses
I've talked about in other videos how negative two to the say even power of
two is different than negative two to the second power this equals positive
four and this equals negative 4 because exponents may act on what they're
sitting right above so because there is a set of parentheses around this entire
fraction and a fraction part itself acts as a grouping symbol we're going to keep
this negative Y in the parenthesis and have negative Y raised to the now within
this denominator this negative is not being affected by this exponent of two
but this negative is inside this parenthesis and there's a power on the
outside of that parenthesis you see I'm not writing a big fraction or a big
set of parentheses around the entire fraction so it's not there's a
parentheses here it's that this big parentheses is coming down and just
wrapping around that that negative line to the second so we have negative Y two
times two is equal to four okay so get this out of the way
we're almost done we have three squared which is equal to nine we have X to the
six and then negative Y to the fourth power or you know negative Y to the 2
times 2 is again fourth power so that's going to be negative Y times negative Y
is positive Y squared and then to the fourth power again which is gonna be
negative Y to the third and then to another power the fourth power it's
going to be positive Y to the fourth if that makes sense as I was saying I
wasn't sure if it was sounded right so we have 9 X to the sixth power over Y to
the fourth I've got four more examples so I'll be right back next to get a
little more complicated as we move along here we have two are the fourth X to the
excuse me third raised to the second power it's just the numerator the how's
that power of two now not the entire fraction over three R squared times X to
the fourth well we need to follow two ER of operations and of course that means
exponents before division so we're going to take that stack exponent situation
here remembering there's only one term inside this grouping symbol and apply
that power to power and let's not forget that two has an exponent 1 so that's
because 2 to the 2nd power are 4 times 2 is 8 and then X to the 3 times 2 which
is the sixth power over 3 R squared X to the fourth
and let's not forget of course you know two squared is equal to four so let's
just change that here because I'm going to run out of space if I don't we have
four thirds we have an R on the top and the bottom where numerator and
denominator we're going to subtract those exponents so we have R to the 8th
minus 2 which is equal to 6 and then we have a base of X in the numerator and
denominator and of course when you divide like bases six minus four is
equal to two so we have four thirds are two to the six x squared make sure to
make some silly mistake that's what my notes say too now here we have a
numerical base that can throw students off and a lot of students want to say 3
divided by 3 is equal to 1 but I don't cancel the ours out and say that it's 1
to the sixth power we have a common base of our and we don't know what the base
is so you're only arithmetic you can see to do is with the exponent which is
correct if you do know what that base is it
doesn't change anything we're going to you know we are dividing like bases so
we're going to copy that common base of 3 and remember that when we divide like
bases we subtract the exponents so it's going to be a plus or minus 2 that comes
out to be 4 minus 2 is equal to 2 and your teacher might be perfectly happy
with that form of the answer however I'm not we have a base with two terms being
added together in the exponent now when do you have exponents you add exponents
and that's what we have here we have addition in the exponent we add
exponents when we multiply like bases so we can take a step further and simplify
it's more by remembering that okay so we must have had or we can rewrite this so
that we have a common base of three and with that addition sign we can rewrite
this as 3 to the K times three squared okay right multiply like bases we can
add those exponents yes that is what we would get so this is an equivalent
expression in l3 squared is equal to nine so it's 9 times 3 to the K power
and I've got 2 more examples so let's get to him
last two well it'll look like a bit of a mess don't they well just take it you
know leave it by little we have exponents you have to do exponent can
you see that that's 3 to the power 3 there you have to do exponents before
you multiply or divide so we're going to apply this power to power so we have
give everything its own exponent and let's attack or do that power to power
with each of those factors and again no addition or subtraction in the numerator
denominators monomial monomial monomial monomial that's when the powered power
will work the monomial one thing ok sue that pains that we have 2 to the 1 times
2 or second power to the second power is 4 so we have 4x to the third to the
second power is X to the sixth we have Y to the fourth to the second power which
is y than 8 times now
negative x squared to the third power let's just take that little bit out
there for saying negative x squared to the third power that is negative x
squared times negative x squared times negative x squared sometimes if you have
a little bit of a question about a piece of a complicated problem like these pull
it out just you know look at it in a tiny little piece negative x squared to
the third power because again we have a negative in there and that can you know
being a little bit difficult this parenthesis is representing that big one
that's around the entire fraction again I know that this base does not have its
own parenthesis but it is negative x squared to the third so negative 1 times
negative 1 is 1 times negative 1 is negative 1 and then X to the second X to
the fourth and now we have X to the sixth so we still do want that negative
sign is being maintained by the fact that it's being raised to an odd power
if this were even the negative we cancel out so we have negative x to the 6 y to
the ninth applying that power power property we're multiplying all the
exponents and then Z to the third okay
well now what do we have we have not much little cancel we can cancel now our
cancel later we have eight whys and then denominator we have nine in the
numerator so we're going to end up with just one Y in the numerator when we're
done but they'll do that at the end four times negative 1 is negative 4 we have X
to the 6 times X to the 6 which is X to the 12 now why am i doing addition and
not multiplying because this X hat or this 6 has its own base of X this six
has its own common base of X and when you multiply like bases you add the
exponents we don't have stacked exponents we're not multiplying without
doing power of power and then there's another or Y to the ninth excuse me over
Y to the 8th over Z or times Z to the 3rd now we have a common base in the
numerator and denominator the wise when you divide like bases probably better if
I had these stacked vertically but when you divide like bases that's what a
fraction bar is Division you subtract the exponents so our final answer is
going to be negative 4 X to the 12th 9 minus 8 is equal to 1 so Y to the first
over Z to the third power here
well a fraction bar is a big grouping symbol and in the numerator I have a
common base of X and when you multiply like bases you add the exponents so this
is going to become X to the k plus 3 plus 2 K minus 1 over X to the K power
all this adds up to K times K is 3 K and 3 minus 1 is 2 so X to the 3k plus 2
over X to the K and now we're dividing like bases now that I've simplified the
numerator as much as possible when you divide like bases you subtract the
exponents so we have X to the 3k plus 2 minus okay and 3k minus 1 K is 2 K so we
have X to the 2k plus 2 a lot of this manipulation we're doing at this point
with these problems you know you're not going to see you know someone do on a
day-to-day basis in their everyday life but this these algebraic skills and
techniques that were built that we're using is building up and building
operating go through you know alpha 1 alpha 2 whatever you're in now into
precalculus and then we're going to get into calculus the first time some of
these techniques you really see getting used in problems that you really can
understand why we care about learning this how it applies to the real life the
calculus will give you the enough power and then algebra along with the idea of
limits and like a babylon and you really see where all this algebra can get tied
into some careers like if you're an engineer or something like that
designing other things that we just happen to use everyday all this math
does apply to real life you know either immediately if you're doing something
like percents or eventually if you're doing all this manipulation of all these
exponents so I'm mr. true let's talk about some negative exponents and power
of 0 in the next video or you can go do your homework yes I
wrote a note about my paper here with all these simplification problems you
know you're done when the coefficients are relatively prime that doesn't mean
that the coefficients are them selves front the coefficients of the numerator
compared to the denominator each base appears only once we had a base here
that showed up twice and I wasn't done until we had that Y only showing up once
and there are no powers of powers like what we had here that we had a
simplified as well now we're done