trig help......?

Question

i^4n+2=? if n is whole numbers


i = imaginary number
please explain how to do it
thanks

i^(4n+2)

i to the power of 4n+2

i^(4n+2)= ? if n is whole numbers

Answer 1

i^(4n+2) = (i^4n)(i^2)
since i^4n = 1 for all whole number values of n, you get
(1)(i^2) = -1
The answer is negative one.
i^1 = sqrt(-1) = i
i^2 = (-1) = -1
i^3 = (-1)sqrt(-1) = -i
i^4 = (-1)(-1) = 1
Then the process repeats
i^5 = (i^4)(i) = (1)(i) = i
i^6 = (i^4)(i^2) = (1)(-1) = -1 ...

Answer 2

If n = 0

i^(4n+2)

i^[4(0) + 2]

i^(0 + 2)

i^2

REMEMBER: i^2 = -1

Substitute -1 for i^2

( -1 )

1( -1)

-1

i^2= -1

Therefore when n= 0, the answer will be -1.

--------------------------------------------------------------------------------

If n = a POSITIVE odd number
For example, n=1,3,5,7,etc.

i^(4n+2)

i^[4(1) + 2]

i^(4 + 2)

i^6

(i^2)^3

REMEMBER: i^2 = -1

Substitute -1 for i^2

( -1 )^3

i^6 = -1

Therefore when n= a POSITIVE odd #, the answer will, ALSO, be -1.

------------------------------------------------------------------------

If n = an POSITIVE even number
For example, n=2,4,6,8,etc.

i^(4n+2)

i^[4(2) + 2]

i^(8 + 2)

i^10

(i^2)^5

REMEMBER: i^2 = -1

Substitute -1 for i^2

( -1 )^5

i^10 = -1

Therefore when n= an even #, the answer will be, once more,-1.

--------------------------------------------------------------------------------

If n = an NEGATIVE odd number
For example, n= -1,-3,-5,etc.

i^(4n+2)

i^[4(-1) + 2]

i^(-4 + 2)

i^(-2)

(i^2)^(-1)

REMEMBER: i^2 = -1

Substitute -1 for i^2

( -1 )^(-1)

(-1/1)^(-1)

(1/-1)^(1)

(-1)^1

-1

Therefore, when n= NEGATIVE odd #, again, the answer is -1.

--------------------------------------------------------------------------------

If n = an NEGATIVE even number
For example, n= -2,-4,-6,-8,etc.

i^(4n+2)

i^[4(-2) + 2]

i^(-8 + 2)

i^(-6)

(i^6)^(-1)

[(i^2)^3]^(-1)

REMEMBER: i^2 = -1

Substitute -1 for i^2

[( -1 )^ 3]^(-1)

(-1)^(-1)

(-1/1)^(-1)

(1/-1)^(1)

(-1)^1

-1

Therefore, when n= NEGATIVE even # -- YES, you guessed it: the answer is -1.

--------------------------------------------------------------------------------
CONCLUTION:
Regardless if n = an odd or an even number, POSITIVE or NEGATIVE or zero, the result will will ALWAYS be -1.

Answer 3

=(i^4)^n * i^2
=1^n * -1
=1 * -1
=-1