To simplify powers of [tex]\(i\)[/tex], we need to understand the cyclical nature of imaginary unit [tex]\(i\)[/tex]. The powers of [tex]\(i\)[/tex] repeat every four steps:
[tex]\[
i^1 = i
\][/tex]
[tex]\[
i^2 = -1
\][/tex]
[tex]\[
i^3 = -i
\][/tex]
[tex]\[
i^4 = 1
\][/tex]
After [tex]\(i^4\)[/tex], the powers repeat the same pattern:
[tex]\[
i^5 = i
\][/tex]
[tex]\[
i^6 = -1
\][/tex]
[tex]\[
i^7 = -i
\][/tex]
[tex]\[
i^8 = 1
\][/tex]
...and so on.
To simplify higher powers of [tex]\(i\)[/tex], we can use the property of modular arithmetic. Specifically, for any integer [tex]\(n\)[/tex], [tex]\(i^n\)[/tex] can be simplified by finding the remainder when [tex]\(n\)[/tex] is divided by 4. This remainder will tell us where we are in the cycle.
For [tex]\(i^{15}\)[/tex]:
1. Calculate the remainder when 15 is divided by 4:
[tex]\[
15 \div 4 = 3 \quad \text{remainder} \quad 3
\][/tex]
This indicates that [tex]\(i^{15}\)[/tex] corresponds to [tex]\(i^3\)[/tex] in our cycle.
2. Refer to our established cycle:
[tex]\[
i^3 = -i
\][/tex]
Therefore,
[tex]\[
i^{15} = -i
\][/tex]
Thus:
[tex]\[
\boxed{-i}
\][/tex]
