Prove `6i ^50+51 ^41−2i ^19+6i ^60=7i`

2i^4 - i^6 = ?

If i = root -1 ,then 1+i ^2+i ^3−i ^6+i ^8 is equal to -

Show that 6i^(50)+5i^(17)-i^(11)+6i^(28) is an imaginary number.

Find the value of i^(4) + i^(5) + i^(6) + i^(7) .

Which of the following complex numbers is equal to (5 + 12i) − ( 9i^2 − 6i), for i = sqrt(−1) ?

if z = 2 + i + 4i^(2) -6i^(3) then verify that (i) (bar(z^(2)) = (barz)^(2))

A particle P starts from the point z_0=1+2i , where i=sqrt(-1) . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z_1dot From z_1 the particle moves sqrt(2) units in the direction of the vector hat i+ hat j and then it moves through an angle pi/2 in anticlockwise direction on a circle with centre at origin, to reach a point z_2dot The point z_2 is given by (a) 6+7i (b) -7+6i (c) 7+6i (d) -6+7i

Prove that the four points 6 hat i-7 hat j ,16 hat i-19 hat j-4 hat k ,3 hat j-6 hat k and 2 hat i+5 hat j+10 hatk form a tetrahedron in space.

Show that four points whose position vectors are 6 hat i-7 hat j ,\ 16 i-19 hat j-4 hat k ,\ 3 hat i-6 hat k ,\ 2 hat i-5 hat j+10\ hat k are coplanar.

The value of i^2 + i^4 + i^6 + i^8.... upto (2n+1) terms , where i^2 = -1, is equal to: