To determine the simplified form of [tex]\( i^{14} \)[/tex], it's important to understand the pattern of powers of the imaginary unit [tex]\( i \)[/tex]. The powers of [tex]\( i \)[/tex] repeat in cycles of 4:
[tex]\[ i^1 = i \][/tex]
[tex]\[ i^2 = -1 \][/tex]
[tex]\[ i^3 = -i \][/tex]
[tex]\[ i^4 = 1 \][/tex]
[tex]\[ i^5 = i \][/tex]
[tex]\[ i^6 = -1 \][/tex]
[tex]\[ i^7 = -i \][/tex]
[tex]\[ i^8 = 1 \][/tex]
And this pattern continues every four powers: [tex]\( i, -1, -i, 1 \)[/tex].
To simplify [tex]\( i^{14} \)[/tex], we need to determine which part of the cycle it corresponds to. This can be done by finding the remainder when 14 is divided by 4:
[tex]\[ 14 \div 4 = 3 \][/tex] with a remainder of 2.
This reveals that [tex]\( i^{14} \)[/tex] has the same value as [tex]\( i^2 \)[/tex] because the remainder is 2. Knowing from the cycle pattern:
[tex]\[ i^2 = -1 \][/tex]
Therefore, the simplified form of [tex]\( i^{14} \)[/tex] is [tex]\(-1\)[/tex].
So, the correct answer is:
C. [tex]\(-1\)[/tex]
