How can I simplify a prime power of i?

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You don't have enough information, actually, because primes can be 1 more or 3 more than a multiple of 4.

Use this cycle:
i^1 = i
i^2 = -1
i^3 = -1 * i = -i
i^4 = -i * i = 1
i^5 = i^4 * i = i
i^6 = i^4 * i^2 = i^2 = -1
i^7 = i^4 * i^3 = i^3 = -i
i^8 = i^4 * i^4 = 1

So the powers cycle through {i, -1, -i, 1}. Basically, break down the exponent by dividing it by 4 with a remainder.

So, i^53 = i^52 * i, since 52 is the largest multiple of 4 that's under 53.
i^53 = i^52 * i = (i^4)^13 * i = 1 * i = i

You can't do anything with the fact that the exponent is prime, but you can rewrite the exponent as either:
4k
4k+1
4k+2
4k+3...Show more

Basically what you need to do when simplifying a power of i (doesn't have to be a prime number as long as it's an integer) is to find the remainder when that power is divided by 4.

Powers of i work like this:

i^1 = i
i^2 = -1 (by definition)
i^3 = (i^2)i = (-1)i = -i
i^4 = (i^2)(i^2) = (-1)(-1) = 1
i^5 = (i^4) i = 1 i = i
i^6 = (i^4)(i^2) = (1)(-1) = -1
i^7 = (i^4)(i^2)i = (1)(-1)i = -i
i^8 = (i^4)(i^2)(i^2) = (1)(-1)(-1) = 1.

That sequence continues for ever. If the power that i is raised to is a multiple of 4, it simplifies to 1.
If the power that i is raised to is one more than a multiple of 4, it is i. If it two more than a multiple of 4, it is -1 and if it is three more than a multiple of 4 it is -i.

Now when 53 is divided by 4, has a remainder of 1. So i^53 = i^1 = i....Show more

i = √(-1) = i

i² = -1

i³ = -1*√(-1) = -√(-1) = -i

i⁴ = √(-1)*√(-1)*√(-1)*√(-1) = -1 * -1 = 1

and it keeps on going in that pattern as you go up
i, -1, -i, 1, i, -1, -i, 1, i, -1 . . . . . forever

so to find out which one 53 is, you divide 53 by 4

53/4 = 13.25

from this you know it reduces to i

how?

you look only at the decimal portion

and compare it to the decimal portions of these

1/4 = .25
2/4 = .5
3/4 = .75
4/4 = a whole number, so nothing after decimal place

and since it had a decimal of .25, then it has the same value as i¹
see, the 1 from the 1/4 came from the i¹

hope this helps...Show more

a million. something to the 0 (^0) is = to a million. So no remember what's indoors the parenthesis the respond could be a million. 2. The GCF of 12X^4 and 6X^3 is undemanding. all you would be waiting to prefer to do is harm each and each and each and each ingredient down. x^4 has aspects of x^4, x^2, and x^a million x^3 has aspects of x^3 and x^a million comparable ingredient for the coefficients 12 has aspects 12, 6, 4, 3, 2, a million 6 has aspects 6, 3, 2, a million hence the respond is 6x^a million...Show more