# Properties of Exponents Simplifying Quotients Division & Negative Exponents

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 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

## 16 Replies to “Properties of Exponents Simplifying Quotients Division & Negative Exponents”

1. Empty Cloud says:

I'm loving the new shirt and hair cut. I hope you do one of these on logarithms

2. ProfRobBob says:

Thank you:) I have a two part video(s) on properties of logs! Just do a search from the homepage of my YouTube channel. Thank you again for watching.

3. ProfRobBob says:

And THANK YOU for watching it!

4. ProfRobBob says:

When you multiply like bases you add the exponents. The 3k came from k + 2k

5. ProfRobBob says:

I am applying the power rule which applies to the exponents. If you find it a little confusing write the base twice to represent the power of two and then multiply.

6. ProfRobBob says:

Thank you…and thank you for watching and your support.

7. Jake L says:

Thanks Mr. Tarrou! I wish my math teacher was you.

8. Dhope yinyang says:

that Money shirt ON POINT!!! oh ya you're a awesome teacher btw haha 😀

9. Florin C. says:

Great and helpful video again! Thanks! Bam!!!!

10. Alex Defatte says:

I fully intend on watching all 500+ of your videos throughout my math career. I Love Math! It all pertains! Thank You for helping me! These videos are timeless!

11. Aaron Facey says:

I don't know why but whenever I have to study for a test in my Algebra 1 Hnrs class I always end up finding one of your videos. Hope you make it to 100k and you just earned yourself a sub. P.S (I'm still trying to find that money shirt anywhere I go) 🙂

12. Amy Bye says:

Thank you, you're a genius.

13. Jyothi Yalamanchili says:

Would it be fine if in the 4th problem if I put r to the 6th and x to the 2 in the numerator?

14. Kitachi Dragneel loves anime says:

This was very useful, but do you happen to have any videos showing what to do if the bases are different and the exponents are, too?
Like 2 to the 3rd power, over 4 to the 8th power.

15. Yaroslav Luchko says:

Wait but for the last one can't you simplify it even further like you did in another example?

16. Waffle Mania says:

Any of y'all here from strawberry crest