# HOW TO EVALUATE THE GIVEN COMPLEX NUMBERS

How to Evaluate the Given Complex Numbers ?

Here we are going to see some example problems based on evaluating the given complex numbers.

## How to Evaluate the Given Complex Numbers - Questions

Question 1 :

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

(i) z + w

Solution :

z + w  =  (5 − 2i) + (-1+ 3i)

=  (5 - 1) + (-2i + 3i)

=  4 + i

Hence the value of z + w is 4 + i.

(ii) z − i w

Solution :

z - iw  =  (5 − 2i) - i(-1+ 3i)

=  5 − 2i + i - 3i

=  5 − 2i + i - 3(-1)

=  5 − 2i + i + 3

=  (5 + 3) + i(-2 + 1)

=  8 - i

Hence the value of z - iw is 8 - i.

(iii)  2z + 3w

Solution :

2z + 3w =  2(5 − 2i) + 3(-1+ 3i)

=  10 - 4i - 3 + 9i

=  (10 - 3) + i(9 - 4)

=  7 + 5i

Hence the value of 2z + 3w is 7 + 5i.

(iv) z w

Solution :

z w =  (5 − 2i) (-1+ 3i)

=  -5 + 15i + 2i - 6i2

=  -5 + 15i + 2i - 6(-1)

=  (-5 + 6) + i(15 + 2)

=  1 + 17i

Hence the value of zw is 1 + 17i.

(v) z2 + 2zw+ w2

Solution :

 z  =  (5 − 2i)z2  =  (5 − 2i)2   =  52 + (2i)2 - 2(5) (2i)  =  25 + 4i2 - 20i  =  25 + 4(-1) - 20i  =  25 - 4 - 20i  =  21 - 20i w  =  (-1 + 3i)w2  =  (-1 + 3i)2   =  (-1)2 + (3i)2 - 2(-1) (3i)  =  1 + 9i2 + 6i  =  1 + 9(-1) - 6i  =  1 + 9 - 6i  =  10 - 6i

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

=  21 - 20i + 2 + 34i + 10 - 6i

=  (21 + 2 + 10) + i(-20 + 34 - 6)

=  33 + 8i

Hence the value of z2 + 2zw+ w2 is 33 + 8i.

(vi) (z + w)2.

(z + w)2   =  (4 + i)2

=  42 + i2 + 2(4)i

=  16 - 1 + 8i

=  15 + 8i

Hence the value of (z + w)2 is 15 + 8i. After having gone through the stuff given above, we hope that the students would have understood, "How to Evaluate the Given Complex Numbers".

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