Evaluate `underset(zto1)(lim)(z^(1//3)-1)/(z^(1//6)-1)`

Evaluate the following limits:- lim_(z rarr 1)(z^(1/3)-1)/(z^(1/6)-1)

Evaluate |{:(.^(x)C_(1),,.^(x)C_(2),,.^(x)C_(3)),(.^(y)C_(1),,.^(y)C_(2),,.^(y)C_(3)),(.^(z)C_(1),,.^(z)C_(2),,.^(z)C_(3)):}|

If z^2+z+1=0 where z is a complex number, then the value of (z+1/z)^2+(z^2+1/z^2)^2+....+(z^6+1/z^6)^2 is

Let z_(1),z_(2) and z_(3) be three non-zero complex numbers and z_(1) ne z_(2) . If |{:(abs(z_(1)),abs(z_(2)),abs(z_(3))),(abs(z_(2)),abs(z_(3)),abs(z_(1))),(abs(z_(3)),abs(z_(1)),abs(z_(2))):}|=0 , prove that (i) z_(1),z_(2),z_(3) lie on a circle with the centre at origin. (ii) arg(z_(3)/z_(2))=arg((z_(3)-z_(1))/(z_(2)-z_(1)))^(2) .

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0

If arg (z^(1//2))=(1)/(2)arg(z^(2)+barzz^(1//3)), find the value of |z|.

Show that the triangle whose vertices are z_(1)z_(2)z_(3)andz_(1)'z_(2)'z_(3)' are directly similar , if |{:(z_(1),z'_(1),1),(z_(2),z'_(2),1),(z_(3),z'_(3),1):}|=0

If z_(1), z_(2) are 1-i, -2 + 4i respectively, find I_(m)((z_(1)z_(2))/(bar(z_(1)))) .