Let's determine the value of [tex]\(( -i )^5\)[/tex].
Firstly, let's recall that [tex]\(i\)[/tex] is defined as the imaginary unit with the property [tex]\(i^2 = -1\)[/tex].
Now, let's work through the exponentiation step-by-step:
1. Write [tex]\(-i\)[/tex] as a complex number: [tex]\(-i = 0 - i\)[/tex].
2. We need to raise this complex number to the 5th power: [tex]\((-i)^5\)[/tex].
3. When taking powers of imaginary numbers, it's useful to consider their polar form. However, we can also directly multiply:
- [tex]\((-i)^2\)[/tex]:
[tex]\[
(-i)^2 = (-i) \times (-i) = i^2 = -1
\][/tex]
- [tex]\((-i)^3\)[/tex]:
[tex]\[
(-i)^3 = (-i) \times (-1) = i
\][/tex]
- [tex]\((-i)^4\)[/tex]:
[tex]\[
(-i)^4 = (-i)^2 \times (-i)^2 = (-1) \times (-1) = 1
\][/tex]
- [tex]\((-i)^5\)[/tex]:
[tex]\[
(-i)^5 = (-i)^4 \times (-i) = 1 \times (-i) = -i
\][/tex]
Thus, [tex]\((-i)^5 = -i\)[/tex].
Therefore, the value of [tex]\(( -i )^5\)[/tex] corresponds to option A. [tex]\(-i\)[/tex].
So, the correct answer is:
A. [tex]\(-i\)[/tex]
