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

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

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.

And THANK YOU for watching it!

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

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.

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

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

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

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

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!

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) 🙂

Thank you, you're a genius.

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

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.

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

Any of y'all here from strawberry crest