Let i be the imaginary unit, i^2 = -1. Find the value of (1 + i)^10.
Correct: C
We need to calculate (1 + i)^10.
It's often easier to first calculate a smaller power and then raise it to the remaining power.
Let's calculate (1 + i)^2:
(1 + i)^2 = 1^2 + 2(1)(i) + i^2
Since i^2 = -1:
(1 + i)^2 = 1 + 2i - 1 = 2i.
Now we can rewrite (1 + i)^10 as ((1 + i)^2)^5:
(1 + i)^10 = (2i)^5.
Next, we apply the exponent to both the coefficient and the imaginary unit:
(2i)^5 = 2^5 * i^5.
Calculate 2^5:
2^5 = 32.
Calculate i^5. The powers of i follow a cycle of 4:
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
i^5 = i^4 * i = 1 * i = i.
Substitute these values back:
(1 + i)^10 = 32 * i = 32i.