Question 158029
I'm assuming that you are in a trig class, and that you have the unit circle with you. Looking at the unit circle, we see that {{{cos(pi/4)=sqrt(2)/2}}} and {{{sin(pi/4)=sqrt(2)/2}}}




{{{ (sqrt (2) / 2 + (sqrt (2) / 2) i )^4 }}} Start with the given expression



{{{ (cos(pi/4) + sin(pi/4) i )^4 }}} Replace {{{sqrt(2)/2}}} with {{{cos(pi/4)}}} and {{{sin(pi/4)}}}



{{{ cos(4(pi/4)) + sin(4(pi/4)) i }}} Use <a href="http://www.maths.abdn.ac.uk/~igc/tch/eg1006/notes/node100.html">De Moivre's Theorem</a> to expand



{{{ cos(pi) + sin(pi) i }}} Multiply



{{{-1+0*i}}} Take the cosine of {{{pi}}} to get -1 and take the sine of {{{pi}}} to get 0



{{{-1}}} Simplify



So {{{ (sqrt (2) / 2 + (sqrt (2) / 2) i )^4=-1 }}}



So the answer is in standard form {{{a+bi}}} where {{{a=-1}}} and {{{b=0}}}