To simplify [tex]\( i^{10} \)[/tex], we need to understand the properties of the imaginary unit [tex]\( i \)[/tex]. The imaginary unit has the following significant properties:
1. [tex]\( i^1 = i \)[/tex]
2. [tex]\( i^2 = -1 \)[/tex]
3. [tex]\( i^3 = -i \)[/tex]
4. [tex]\( i^4 = 1 \)[/tex]
These powers repeat in cycles of 4. Therefore, any power of [tex]\( i \)[/tex] can be simplified by expressing it in terms of one of these four fundamental values.
To find [tex]\( i^{10} \)[/tex], we need to determine its position within this repeating cycle. We do this by calculating the remainder when 10 is divided by 4:
[tex]\[
10 \div 4 \text{ gives a quotient of 2 and a remainder of } 2.
\][/tex]
The remainder is what determines the simplified form of our original expression. Specifically:
- If the remainder is 0, [tex]\( i^n = i^0 = 1 \)[/tex]
- If the remainder is 1, [tex]\( i^n = i^1 = i \)[/tex]
- If the remainder is 2, [tex]\( i^n = i^2 = -1 \)[/tex]
- If the remainder is 3, [tex]\( i^n = i^3 = -i \)[/tex]
For [tex]\( i^{10} \)[/tex]:
[tex]\[
10 \mod 4 = 2
\][/tex]
Therefore, [tex]\( i^{10} \)[/tex] simplifies to [tex]\( i^2 \)[/tex], which we know is:
[tex]\[
i^2 = -1
\][/tex]
Thus, the simplified form of [tex]\( i^{10} \)[/tex] is:
[tex]\[
-1
\][/tex]
Hence, the correct answer is [tex]\(-1\)[/tex].
