Apply Euler's relation
e^(itheta)=costheta+isinthetaeiθ=cosθ+isinθ
e^(11/12ipi)=cos(11/12pi)+isin(11/12pi)e1112iπ=cos(1112π)+isin(1112π)
e^(1/4ipi)=cos(1/4pi)+isin(1/4pi)e14iπ=cos(14π)+isin(14π)
i^2=-1i2=−1
Therefore,
e^(11/12ipi)*e^(1/4ipi)=(cos(11/12pi)+isin(11/12pi))*(cos(1/4pi)+isin(1/4pi))e1112iπ⋅e14iπ=(cos(1112π)+isin(1112π))⋅(cos(14π)+isin(14π))
=cos(11/12pi)cos(1/4pi)-sin(11/12pi)sin(1/4pi)+i(cos(11/12pi)sin(1/4pi)+sin(11/12pi)cos(1/4pi)))=cos(1112π)cos(14π)−sin(1112π)sin(14π)+i(cos(1112π)sin(14π)+sin(1112π)cos(14π)))
=cos(11/12pi+1/4pi)+isin(11/12pi+1/4pi)=cos(1112π+14π)+isin(1112π+14π)
=cos(14/12pi)+isin(14/12pi)=cos(1412π)+isin(1412π)
=cos(7/6pi)+isin(7/6pi)=cos(76π)+isin(76π)
=-sqrt3/2-1/2i=−√32−12i