Sure, let's break this down step by step.
We need to determine the value of [tex]\(i^{34}\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit. The powers of [tex]\(i\)[/tex] follow a cyclic pattern every four exponents:
1. [tex]\(i^1 = i\)[/tex]
2. [tex]\(i^2 = -1\)[/tex]
3. [tex]\(i^3 = -i\)[/tex]
4. [tex]\(i^4 = 1\)[/tex]
This cycle repeats every four exponents. Therefore, to find [tex]\(i^{34}\)[/tex], we need to determine where 34 falls within this cycle. We can do this by finding the remainder when 34 is divided by 4.
[tex]\[ 34 \div 4 = 8 \text{ remainder } 2 \][/tex]
So, the remainder is 2, which means [tex]\(i^{34}\)[/tex] is equivalent to [tex]\(i^2\)[/tex] based on the cycle.
Now, looking at the cycle, we know that:
[tex]\[ i^2 = -1 \][/tex]
Thus, the value of [tex]\(i^{34}\)[/tex] is [tex]\(-1\)[/tex].
So, the correct answer is:
A. [tex]\(-1\)[/tex]
