2/(1+i)^4
(1+i)^2 = 1 + 2 i - 1 = 2 i
so
(1+i)^4 = -4
2/-4 = -1/2
1+i^4 = 1+4i+6i^2+4i^3+i^4
= 1+4i-6-4i+1
= -4
or,
1+i = √2 cis 3π/4
so, (1+i)^4 = 4 cis3π = -4
I guess you can take it from there, right?
To find the value of 2/(1+i)^4, we need to simplify the expression using the laws of complex numbers.
First, let's simplify the denominator, (1+i)^4:
To find the fourth power of (1+i), we can use the formula (a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.
In this case, a = 1 and b = i. Plugging these values into the formula, we get:
(1+i)^4 = 1^4 + 4(1^3)(i) + 6(1^2)(i^2) + 4(1)(i^3) + i^4
= 1 + 4i + 6(-1) + 4(-i) + 1
= 1 + 4i - 6 + (-4i) + 1
= -4.
Now, let's substitute this value back into the original expression:
2/(1+i)^4 = 2/(-4).
Finally, we simplify further:
2/(-4) = -1/2.
Therefore, 2/(1+i)^4 simplifies to -1/2.