Question

evaluate: 1 + i^10 + i^1000
​

Answer 1

$1 + i^{10} + i^{1000}\\ \\1 + (i^{2})^{5} + (i^{2})^{500} \\ \\we know ------------ : i^{2} = -1\\ \\1 + (i^{2})^{5} + (i^{2})^{500} \\\\1 + (-1)^{5} + (-1)^{500}$

for odd power (-1) remains negative

but for even power (-1) becomes positive

So,

$1 + (-1)^{5} + (-1)^{500} \\\\ 1 - 1 + 1\\\\= 1$