HOW TO EVALUATE THE GIVEN COMPLEX NUMBERS

Evaluate the following if z = 5 - 2i and w = -1 + 3i.

Example 1 :

z + w

Solution :

z + w  =  (5 - 2i) + (-1+ 3i)

  =  (5 - 1) + (-2i + 3i)

  =  4 + i

Example 2 :

z − iw

Solution :

z - iw  =  (5 − 2i) - i(-1+ 3i)

  =  5 − 2i + i - 3i² 

=  5 − 2i + i - 3(-1) 

=  5 − 2i + i + 3

=  (5 + 3) + i(-2 + 1)

=  8 - i

Example 3 :

2z + 3w

Solution :

2z + 3w =  2(5 − 2i) + 3(-1+ 3i)

=  10 - 4i - 3 + 9i

=  (10 - 3) + i(9 - 4)

=  7 + 5i

Example 4 :

zw

Solution :

zw =  (5 − 2i) (-1+ 3i)

=  -5 + 15i + 2i - 6i²

=  -5 + 15i + 2i - 6(-1)

=  (-5 + 6) + i(15 + 2)

=  1 + 17i

Example 5 :

z² + 2zw+ w²

Solution :

z² + 2zw+ w²  =  21 - 20i + 2 + 2( 1 + 17i) + 10 - 6i

  =  21 - 20i + 2 + 34i + 10 - 6i

  =  (21 + 2 + 10) + i(-20 + 34 - 6)

  =  33 + 8i

Example 6 :

(z + w)²

(z + w)²   =  (4 + i)²

  =  4² + i² + 2(4)i

   =  16 - 1 + 8i

  =  15 + 8i

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