66.

Find the square root of the following complex numbers :

-5 + 12i

Let,  = x + iy                                                              ....(i)

Squaring both sides, we get

-5 + 12i = (x + iy)²

   - 5 + 12i = x² + i²y² + 2xyi

- 5 + 12i = x² - y² + 2xyi                                                               ....(ii)

Equating real and uimaginary parts, we get

            x² - y²  = -5                                                                        ....(iii)
              2xy = 12                                                                            ....(iv)

      4x²y² = 144 

Also,                    (x²+ y²)² = ((x² - y²)² + 4x² y²

                     (x²+ y²)² = 25 + 144 = 169

                      x²- y²  = 13                                                            ...(v)            

                                                  (∵ x²- y² cannot be negative for x, y  R)
From (iii) a nd (v), we have

                    x²- y²  = -5
                
                     x²+ y² = 13
Adding, we get      2x² = 8 x² = 4 x = 2
When x = 2, from (iii), 2(2) (y) = 12 or y = 3
When x = 2, from (iii), 2(-2)y = 12 or y = -3

Putting in (i)  we have

or       

Hence,