Calculation of complex exponentiation

Question

I understand that:

$a^{ib}$ = $Cos[bLn(a)]+iSin[bLn(a)]$

Mathematica calculates N[$7^{6i}$] = $0 + 117649i$ $\quad$ and $\quad$ N[Cos[6Log[7]] + iSin[6Log[7]]] = $0.628704848492136 - 0.7776440146252527i$

I would expect '$0 + 117649i$' and '$0.628704848492136 - 0.7776440146252527i$' to differ by a period of $\pi$, but that doesn't seem to be the case.

That is, 0 $\neq$ 0.628704848492136 $\pm$ $n\pi$

What am I not understanding correctly? Thank you.

Answer 1

Your expectation is wrong because you are not respecting operator precedence:

N[7^6i] is interpreted a $N[(7^6)i]= (7^6) i = 117649 i$

If you use $N[7^{6i}] =$ N[7^(6i)] you get the coinciding result

$ N[7^{6i}] = \cos(6 \ln 7)+i\sin(6\ln 7)=0.6287048484921367- 0.7776440146252522 i$