HOW TO EXPAND COMPLEX NUMBERS

How to Expand Complex Numbers ?

Here we are going to see, how to expand complex numbers.

How to Expand Complex Numbers - Examples

Write the following expression in the form a + bi, where a and b are real numbers.

Question 1 :

(3 + 4i)2

Solution :

The given question exactly matches the algebraic identity

(a + b)=  a2 + 2ab + b2

(3 + 4i)2  =  32 + 2(3)(4i) + (4i)2

 =  9 + 24i + 16(-1)

 =  -7 + 24i

Question 2 :

(5 - 2i)2

Solution :

The given question exactly matches the algebraic identity

(a - b)2  =  a2 - 2ab + b2

(5 - 2i)2  =  52 + 2(5)(-2i) + (-2i)2

 =  25 - 20i - 4

 =  25 - 4 - 20i

 =  21 - 20i

Question 3 :

(4 - 7i)2

Solution :

The given question exactly matches the algebraic identity

(a - b)2  =  a2 - 2ab + b2

(4 - 7i)2  =  42 - 2(4)(7i) + (7i)2

 =  16 - 56i + 49i2

 =  16 - 56i + 49(-1)

 =  16 - 56i - 49

=  -33 - 56i

Question 4 :

(5 + √6i)2

Solution :

The given question exactly matches the algebraic identity

(a - b)2  =  a2 - 2ab + b2

(5 + √6i)2  =  52 - 2(5)(√6i) + (√6i)2

 =  25 - 10√6i + 6i2

 =  25 - 10√6i - 6

 =  19 - 10√6i

Question 5 :

(1 + √3i)3

Solution :

The given question exactly matches the algebraic identity

(a + b)3  =  a3 + 3a2 b + 3ab2 + b3

(1 + √3i)3  =  13 + 3(1)2 √3 + 3(1)√32 + √33

 =  13 + 3√3 + 3(3) + 3√3

 =  1 + 6√3 + 9

=  10 + 6√3

Question 6 :

[(1/2) - (√3/2)i]3

Solution :

The given question exactly matches the algebraic identity

(a - b)3  =  a3 - 3a2 b + 3ab2 - b3

[(1/2) - (√3/2)i]3  

=  (1/2)3 - 3(1/2)2 (√3i/2) + 3(1/2)(√3i/2)2 + (√3i/2)3

=  (1/8) - (3√3i/8) - (9/8) - (3√3i/8)

=  (-8/8) - (6√3i/8)

=  -1 - (3√3i/4)

After having gone through the stuff given above, we hope that the students would have understood "How to Expand Complex Numbers". 

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