How to Expand Complex Numbers ?
Here we are going to see, 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)
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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