Answer :

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].