Calculating i Raised to Arbitrary Exponents Calculating i raised to arbitrarily high exponents
Calculating i Raised to Arbitrary Exponents
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- Now that we've seen that as we take i to higher and higher powers it cycles between 1, i, negative 1, negative i, then
- back to 1, i, negative 1 and negative i. I wanna see if we can tackle some -- I guess you could call them
- trickier problems -- and you right see them in this surface and they are also a kind of fun to do them
- and realise that you can use this the fact that I the powers of I cycle through these values you can use this
- to really, on the back of an envelop, take arbitrarily high powers of i. So let's try, just for fun,
- let's see what i to one hundredth power is and the realisation here is that 100 is a mutiple of 4, so you could
- say this is the same thing as i to the, i to the 4 times 25th power and this is the same thing,
- just from our exponent properties i to the 4th power raised to the 25th power, right, if you have
- something raised on an exponent and that is raised to an exponent that's the same
- thing as multiplying the two exponents and we know that i to the 4th, that's pretty
- straightforward; i to the 4th is just 1, i to the 4th is 1 so this is 1 so this is equal to
- the 1 to the 25th power which is just equal to, just equal to 1. So once again we use
- this kind of cycling ability of i when it takes its powers to figure out very high
- exponent of i. Now let's say we try something a little bit stranger, lets try i to the,
- let's try it i to the five hundred and first power. Now in this situation 501 is not a
- multiple of 4 so you can't just do that simply but what you could do is you could write
- this is a product of two numbers; one that is a multiple, one that is i to a multiple
- of 4th power and then one that isn't, also you could rewrite this: 500 is, is a multiple of
- 4, so you could write this is i to the five hundredth power, i to the five hundredth
- power, times i to the first power, right? you have same base, when you multiply
- them you can add exponents, so this is i to the five hundred and first power. And we
- know that this is the same thing as i to ... i to five hundred power is the same thing as i to
- the fourth power four times what? Four times one hundred and twenty-five is 500, so
- that's this part right over here: i to the 500 is the same thing as i to the 4th to the
- 125th power and then that times i to the first power, times i to the first. Well i to the
- 4th is 1, 1 to the 125th power is just going to be 1, this whole thing is 1 and so we
- are just left with, we are just left with i to the first, so this is going to be equal to i. So
- it seems really daunting problem, something that you have to sit and do all day
- but you can use the cycling result; i to the 500 is just going to be 1 and so i to the
- 501 is just going to be i just times that. So i to any multiple of 4 -- let me write this
- generally -- so if you have i to any multiple of 4, so this right over here is, well we'll just
- restrict k to be non negative right now; k is greater or equal to zero
- so if we have i to any mutiple of 4 right over here we are going to get, we are going to get
- 1 because this is the same thing as i to the fourth power to the kth power and
- that is the same thing as 1 to the kth power which is clearly equal to 1 and if we have
- anything else, if we have i to the fourth k plus 1 power or plus 2 power we can then just
- do this technique right over here. So let's try with a few more problems, just to
- make it clear that you can really really arbitrarily crazy things. So let's take i to the
- seven thousand three hundred and twenty-first power. Now we just have to figure out
- -- this is going to be some multiple of 4 plus something else, so to do that, well you
- could just do that by sight 7320 is dividable by 4, you can verify that by hand and then you have one left
- over also this is going to be i to the 7320 times i to the first power, this is a multiple
- of 4 this right here is just a multiple of 4, I know that because any 100 is a multiple of
- 4, any 1000 is a multiple of 4 and any 100 is a multiple of 4 and the 20 is a multiple of
- 4 and so this right over here simplified to 1... sorry, that's not ??? to the ith power either the first power, 7321 is 7320 plus
- 1 and so this part right over here is going to simplify 1 and its going to be left
- with i to the first power or just i. Let's do another one. i to the, i to the ninety, ninety,
- ninety, let me try something interesting, i to the ninety-ninth, i to the ninety-ninth power.
- So once again, what's the highest multiple of 4 that is less than 99? It is 96, it is 96,
- so this is same thing as the i to the 96th power times i to the 3th power, right? If you
- multiply these same base, add the exponent, you'll get i to the 99th power, i to the
- 96th power, so this is a multiple of 4, this is i to the 4th and then that to the 16th
- power, so that's just 1 in the 16, so this is just 1 and then you'll just lift with i to the
- third power and you could either remember that i to the 3th power is equal to... you
- could remember its equal to negative i or if you forget that you could just say look
- this is just samething as i squared times i, this is equal to i squared times i, i
- squared by definition is equal to negative 1 so you have negative 1 times i, is equal
- to i, is equal to negative i. Let me do one more, just for, just for the fun of it. Let's take i
- to the 38th power. Well once again this is equal to the i to the 36th times i squared, I'm
- doing i to the 36th power just largest multiple 4 that goes into 38 what's left over is 2
- this simplifies to 1 and I'm just left with i squared which is equal to negative 1.
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