this is not the time to sing only you but rather we have to sing only i ladies and
gentlemen have a look right here starting with i and let's look at this as i to the first power
and everybody knows 4 over four is equal to one but i know one is the same as four over four
so i'm gonna put this down as i raised to the four over four power well let's continue by
looking at this as i to the fourth power and then raised to the one over four power well i to the
fourth power is just equal to one and then this right here let's just put down the fourth root
and finally we can see that the fourth root of one is of course just equal to one right
well from here to here i have no problem with it but if you look back what we are trying to say
is that i is equal to one oh my god what kind of new discovery is this i have never knew about this
seriously i is equal to one no of course not and you guys told me this already the mistake
is at step number two right here right well in this video let me explain all the things that
i think you should know first from here to here again we have no problem with the fourth rule of
one is equal to one but we just cannot say one is equal to i so let me cross this out first and now
let's just look back is it okay for us to say i is equal to the fourth root of one or maybe look
at this backwards the fourth root of one is equal to i what do you guys think well the truth is i
will tell you this right here is somewhat okay so let me just put down okay but in a quotation mark
in fact this is what i learned after i came to the united states it seems that you can save
whatever you want to say as long as you put a quotation mark around it now where people
seems to be you know they can understand you automatically but anyway this is somewhat okay and
the reason is because they are in fact four fourth roots so let me just write this down in words
of one
and especially when we're dealing with complex numbers when we're talking about the first root
usually we shall find out all the possibilities and the secret is that in fact i is one of them
and here let me just tell you guys what are the four first truths
well we will have one negative one i and also negative i and of course i is one of them
so now you see if you put this down what are you trying to say first root of one is equal to i
you only put down one possibility hm how about the other four right how about the other three i mean
this is like what you know yes i tell i'm telling you hey i'm going to treat you a
full course meal tonight but in the end i'm just buying you the soup and
you have to pay for the entree dessert the drink the second course things like that
how would you feel so i'm going to leave that to you guys okay for that all right so now
let's really talk about this right here why are we doing this this way could we have done it as i to
the one over fourth power first and then raise to the fourth power that's the main issue that
we would like to talk about right here so let's go ahead and put that down if we only put down i
to the fourth power and then raise to the one over fourth power versus i to the one over four first
and then raised to the fourth power well which one is correct or are they actually the same
here let me tell you in fact this right here will actually give you four answers
just like what we have been talking about right here and here you actually will just get one
answer the truth is whenever we are working with a fractional exponent
with the complex numbers we have to be super super careful and now let me just explain to you
guys why these are and how they are right here now let's take a look at what do we do when we have i
to the fourth power and then raise to the one over fourth power well we are going to do this inside
out and we will find out the possibility check this out when we have this of course i to the
fourth power is just equal to one and then we are going to raise this to the one over fourth power
we are going to do everything in the polar form and then let's see to do so i'm going to take one
right put on the complex languages right here this is the real axis and this is the imaginary axis r
is equal to 1 and then the theta is just 2 um pi so far so good so we are going to write the one
in terms of the polar form right so this right here will be 1 times e to the i theta which is
2 um pi like so and then after that we will you know raise this to the one over fourth power and
now you see we have this raised to the one over fourth power we will actually get four answers
to do so what we can do is multiply the powers we get e to the i and then we just have n over 2
pi and to get all the answers all we have to do is plug in 0 1 2 and 3 into the n
and the truth is this right here is equal to 1 and this right here is just equal to i
and this right here is equal to negative 1 and this right here is equal to negative i as we can
see we do have four different answers okay so that was good now let's check this out what
if we start off by looking at i to the one over fourth first and then raised to the [Music] fourth
power like that well this time though because we have the one over four right here for the i we
are going to take i and put that into the polar phone first and then there will be four answers
here this is i and the polar form is going to be the following i is right here and again let's just
make it prettier r e and then i am anyway r is equal to one and then the angle theta is equal to
that's pi over 2 and don't forget just add the 2 pi so for i this is what we are going to get
r is 1 so we just have e and then i and let's get the common denominator so we'll just get 4 n
plus 1 over 2 right and then we also have the pi right here so this is what the i is
then we are going to raise this to the one over fourth power
and then we are going to raise this to the
fourth power all right so that's what we have now check this out what we are going to do is we can
do this times that and then remember we have four answers so just make sure you plug in n is
equal to zero one two and three and let's just go ahead and do that right here i will just do
two things at one time and it's equal to zero one two and three so what exactly are we going to get
for this part when n is equal to zero this is just one half pi times one over four is just one over
eight and pi also the i so the first answer here is e to the i pi over eight next we will have e
and then when you plug in one you get five and then still have the eight on the bottom so
i five pi over eight and then e to the next one you get nine so i nine pi over eight and lastly
e to the 13 right so that's i and then 13 pi over 8. pretty good so
this part the inside we do have four answers however we are going to take each one and then
raise to the power of four and now here's the deal we're going to just multiply the exponents
it still looks like we have four different answers right but you know what let me tell you guys that
the truth is this right here is equal to i this right here 5 pi over 2 which is also that angle
it's also i likewise 9 pi over 2 it's also right there is i and lastly
is 3i no just kidding i don't know why i wanted to do that but anyway i'm sorry i
so this is not the time to think only you but rather we have to sing only i
i know it sounds so conceited but uh no in math it's okay the answer is just only i
we don't have my answer so you see this is actually a legitimate way to take care of that
uh fractional situation and you might be wondering seriously why is that when we do the exponents in
a different order how can we end up with different results well you can bring on the complex numbers
and let me tell you though in fact the problem the real problem with this is that we didn't reduce
we didn't reduce the fraction so instead of looking at i to the four over four
like this for that you get two different answers but if you look at let's say i to
this three over four which the fraction is in the reduced form already well you end up with
the same answer regardless which way you do it so now let me show you guys that real quick
what if we do i and then do the third power first and then raise to the one over fourth power
in this case you see three over four you cannot reduce the fraction for that well anyway inside
wise this is going to be negative i and then we are going to look at this raise to the one over
fourth power and you know the usual business let's take the i to the let's take the negative i now
to the complex plank which is right here this time though the angle is uh three pi over two and then
you just add a two m pi right okay and then again let's put on re and also the i m because i made a
horrible mistake before so i really feel like i should put down r e and also the i m if you guys
know what i'm talking about thank you anyway so negative i is going to be well r e i theta so r
is equal to 1 and then we have e and then i and let's just get the common denominator so we have
4n and then plus 3 over 2 and then the pi so this is what we have for negative i and we are going to
raise this to the 1 over 4 power well again we can just plug in 0 1 2 and 3 and also multiply
the exponents so let's go ahead and do that real quick and perhaps i will multiply the exponent
first it's slightly serious thing anyway this time we get 4 m plus three over eight and then pi
and then n is equal to zero one two three plugging plugging plugging so we do have these four answers
that's it now have a look right here half half half a look right here suppose today
we did that by looking at i to the one over four first and then raised to the third power
well i to the one over four is equal to what we talked about earlier so i'm just going to
put down the result now we are going to raise everybody right individually to the third power
i'll do the same yes they are this checks with that right so check check and then
we have the 15 so this right here so checks so two checks two checks now here's the deal
27 pi over eight is over 2 pi already so what you have to do is just subtract
2 pi and then give you a common denominator so this is actually the same as 11 pi over eight
that's pretty much the period of sine cosine so this right here is three checks one two three
this is actually the same as that one two three right so again this is the same as e to the
i 11 pi over 8. and lastly if you look at 39 pi over 8
you have to subtract multiple of 2 pi and just make sure that the angle is within 0 to 2 pi
well in that case you are going to subtract 4 pi and that will give you 7 pi over 8.
so this guy 1 2 3 4 is actually the same as 1 2 3 4. this is the same as e to the i seven pi over
eight so ladies and gentlemen as you can see when you have this right here in fact it doesn't matter
how you do it so when exactly can we do that well let me summarize all this for you guys
well let's say z is a complex number and then we have m over n so this right here
okay is equal to either z to the 1 over n and then to the nth power
and you can also define this to be z to the nth power and then to the one over this power
so they are okay under the assumption that if the greatest common divisor of m and n is equal to one
if the m and m they don't have any common factors other than one
then you can do either way just what we are doing right here but suppose this is reducible come on
just go ahead and reduce the fraction if you ever have to do that i to the six over eight well
in this case seriously just go ahead and do
i to the three over four and then be happy come on all right if g c t of m n is not equal to 1 then
just reduce the fraction and over n all right or you do it this way the right way to deal with the
fractional exponent is that you do z to the 1 over n first and then raised to the m's power
so this shows you why earlier we should do i to the 1 over 4 first and then to the fourth power
right so hopefully this helps if you find this video interesting then you can continue the
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so i think that was the reason why you guys told me this right here is the
mistake right but you know what in my opinion this right here i think it actually has a
bigger issue and of course i'm putting this in a quotation mark because why seriously why
why in the world would you like to write i to the first power as i to the 4 over 4
it doesn't do us any good and then it becomes so complicated after that oh my god seriously
yeah so in my opinion that's where the bigger issue is but let me tell you though if you really
want to deal with i to the 4 over 4 just like what we talked about earlier be sure you do the
1 over 4 power first and then do the fourth power on the other side
or better you just write it as i to the first power man who did that yes it was me