Algebra Question

Question

Determine the truth values for the following statements. Please provide a clear motivation.

a) 2+(-i)⁵ is a real number

Answer 1

Note that the definition of i is that it is the number such that i=√-1. This means that i²=-1. Squaring both sides means i⁴=1. Multiplying both sides by i then implies i⁵=i.

So to figure out (-i)^5 first notice that

-i = -1*i

=> (-i)⁵ = (-1*i)⁵ = (-1)⁵ * (i)^5

Here you should notice that -1 to an odd exponent is always -1 and i⁵=i. So

(-1)^5 * (i^5) = -1*i = -i

So finally 2+(-i)^5 = 2+(-i) which is not a real number. The answer is false.