Can you please help me to understand why the first line is not correct? - $$i^{15}=(i^4)^\frac {15}4=1$$ $$i^{15}=i^{12}\cdot i^3=-i$$
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1See https://math.stackexchange.com/q/438/42969 or https://math.stackexchange.com/q/3219025/42969 – Martin R Mar 16 '23 at 09:02
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$i^{15}$ is determined by working out $15 \equiv 3(\mod4)$. You then work out $i^3$, which is $-i$. The reason you got the wrong result on the first line is because the fourth root of any number has $4$ solutions, the positive real ($1$ in this case), the negative real ($-1$ in this case), the positive imaginary ($i$ in this case) and the negative imaginary ($-i$ in this case). The one that is usually indicated using the fourth root is the positive real, but in this case, to get the correct answer, you would have to use the positive imaginary.
