Question 1152858
<br>
{{{(1+ic)^6}}}<br>
{{{(1^6)+6(1^5)(ic)+15(1^4)((ic)^2)+20(1^3)((ic)^3)+15(1^2)((ic)^4)+6(1^1)((ic)^5)+((ic)^6)}}}<br>
{{{1 + 6ic - 15c^2 - 20ic^3 + 15c^4 + 6ic^5 - c^6}}}<br>
{{{(1-15c^2+15c^4) + (6c-20c^3+6c^5)i}}}<br>
We want the imaginary part to be 0:<br>
{{{6c-20c^3+6c^5 = 0}}}
{{{2c(3-10c^2+3c^4) = 0}}}
{{{2c(3-c^2)(1-3c^2) = 0}}}<br>
{{{c= 0}}} OR {{{c^2 = 3}}} OR {{{c^2 = 1/3}}}<br>
The 5 values of c for which the expression is real are
{{{0}}}
{{{sqrt(3)}}} and {{{-sqrt(3)}}}
{{{sqrt(1/3)}}} and {{{-sqrt(1/3)}}}<br>