To simplify [tex]\( i^{38} \)[/tex], we can use properties of the imaginary unit [tex]\( i \)[/tex]. Recall that:
[tex]\[
i = \sqrt{-1}, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
\][/tex]
The powers of [tex]\( i \)[/tex] cycle every 4 terms:
[tex]\[
i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad i^5 = i, \quad \text{and so on}
\][/tex]
Given that every 4th power of [tex]\( i \)[/tex] returns to 1, we can reduce the exponent modulo 4 to simplify our calculation. Since we are looking to simplify [tex]\( i^{38} \)[/tex]:
1. Compute the exponent modulo 4:
[tex]\[
38 \mod 4 = 2
\][/tex]
2. This means [tex]\( i^{38} \)[/tex] simplifies to [tex]\( i^2 \)[/tex]:
[tex]\[
i^{38} = i^{2}
\][/tex]
3. From the properties of [tex]\( i \)[/tex], we know:
[tex]\[
i^2 = -1
\][/tex]
Thus, the simplified form of [tex]\( i^{38} \)[/tex] is:
[tex]\[
-1
\][/tex]
Therefore, [tex]\( i^{38} = -1 \)[/tex].
