Question 195121
Are you sure that it's not a square root and the expression is {{{(1/2-(sqrt(3)/2)*i)^3}}}??? If so, then...



Take note that {{{cos(60)=1/2}}} and {{{sin(60)=sqrt(3)/2}}}. So this means that




{{{(1/2-(sqrt(3)/2)*i)^3}}} Start with the given expression.



{{{(cos(60)-sin(60)*i)^3}}} Replace {{{1/2}}} with {{{cos(60)}}}. Replace {{{sqrt(3)/2}}} with {{{sin(60)}}} (see above)



{{{cos(60*3)-sin(60*3)*i}}} Use <a href="http://en.wikipedia.org/wiki/De_Moivre's_formula">De Moivre's formula</a> to expand.



{{{cos(180)-sin(180)*i}}} Multiply



{{{-1-0*i}}} Evaluate the trig functions.



{{{-1}}} Simplify



So {{{(1/2-(sqrt(3)/2)*i)^3=-1}}}