Question

What is the simplest form of i^16 plus i^6 minus 2i^5 plus i^13?

Answer 1

$\bf \textit{recall that}\qquad i^4=1\qquad i^3=-i\qquad i^2=-1\\\\ -------------------------------\\\\ i^{16}+i^6-2i^5+i^{13}\implies i^{4\cdot 4}~+~i^{4+2}~-~2i^{3+2}~+~i^{4+4+4+1} \\\\\\ (i^4)^4~+~i^4\cdot i^2~-~2\cdot i^3\cdot i^2~+~i^4\cdot i^4\cdot i^4\cdot i^1 \\\\\\ (1)^4+[1\cdot -1]-[2(-i)(-1)]+[(1)(1)(1)(i)] \\\\\\ 1-1-2i+i\implies -i$