quick math help, plz!!complex numbers?

Question

Hello, this is a hard question, I can't answer it so need help Plz help

for wht integrl vlues of a is (a+i)^4 also an intger?

i=I think it means complex, or imaginary number

Answer 1

You mean "for what integral values of a is (a+i)⁴ also an integer?" This isn't sms, you can afford to type complete words on here.

Anyway, first, we must compute what (a+i)⁴ actually is:

(a+i)⁴
((a+i)²)²
(a²-1 + 2ai)²
(a⁴ - 2a² + 1 - 4a²) + (4a³ - 4a)i
a⁴ - 6a² + 1 + (4a³ - 4a)i

Now, obviously the real part will always be an integer whenever a is, so the only remaining requirement to check is that the imaginary part is zero. Thus the integer values of a for which (a+i)⁴ will be an integer are precisely the integer values of a for which 4a³ - 4a = 0, and that happens when:

4a³ - 4a = 0
a³ - a = 0
(a²-1)a = 0
(a-1)(a+1)a = 0
a=0 or a=-1 or a=1.

So the solutions are -1, 0, and 1. And we are done.

Answer 2

(a + i)^4 = a^4 + 4a^3 i + 6a^2 i^2 + 4a i^3 + 1
= a^4 + 4a^3 i - 6a^2 - 4a i + 1. (using i^2 = -1)

This will be an integer if and only if
4a^3 i - 4a i = 0 -> 4(a^3 - a) = 0.

Solve the equation for a.

Answer 3

a = 1, -1

Addendum: Okay, 0 too. Give the other guy his 10 points.