I'm answering an MCQ regarding the topic. Some of my answers to the questions doesn't match the answer key. (It has no solution only the final answer.) I just want to know whether I'm right or wrong or maybe the key to corrections is erroneous. Just a yes or no is suffice or a tip if its wrong.
1.) Simplify the expression $i^{1999} + i^{1999}$ where I is an imaginary number.
- $i^{1996} \cdot i^3 + i^{1996} \cdot i^3$
- $i^{1996}$ cancels out as it is equal to 1, so: $i^3 + i^{3} =$ -2i
However, the answer key states that the correct answer is 0.
2.) Evaluate $4i^{410} - i^{864} + i^{601} + i^{1203}$
- $4(i^{408} \cdot i^{2}) - i^{864} + i^{600} \cdot i^1 + i^{1200} \cdot i^3$
- Those with exponents divisible by 4 will be equal to 1, so : $4i^2 - 1 + i + i^3 = $ -5
But the answer key states that the correct answer should be 3.
3.) If $i^2 = -1$, then $i^7 - i^6 + i^5 = $ ?
- $i^4 \cdot i^3 - i^4 \cdot i^2 + i^4 \cdot i^1$
- $i^4$ is equal to 1 so: $i^3 - i^2 + i = $ 1
But the answer key states that it should be -i
So, is there something wrong on how I answer this kind of questions? Thank you in advance. :)
I get
$i^{1999}=-i$, so $i^{1999}+i^{1999}=-2i$.
$4i^{410} - i^{864} + i^{601} + i^{1203} =-4-1+i-i=-5$.
$i^7-i^6+i^5=-i+1+i=1$.