HOW TO EXPAND COMPLEX NUMBERS

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)²  =  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)

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.