1: Express the given complex number in the form a + ib: (5i)(-3/5 i)
Solution:
(5i)(-3/5 i) = -5 x 3/5 x i x i
= -3i2
= -3(-1) = 3
2: Express the given complex number in the form a + ib: i9 + i 19
Solution:
i9 + i 19 = i4×2+1 + i4×4+3
= (i4)2.i + (i4)4.i3
[i4 = 1 and i3 = -1]
= 1 x i+1 x (-i)
= i + (-i)
= 0
- Express the given complex number on the form a + ib: i-39
Solution:
i-39 = i-4×9-3
=(i4)-9 x i-3
= (1)-9 x i-3
= 1/i3
= 1/-I x i/i = -i/i2 x i/i = -i/i2 = -i/-i = i
- Express the given complex number in the form a + ib: 3(7 + 7i)+i(7+7i)
Solution:
3(7 + 7i)+i(7+7i) = 21 + 21i + 7i +7i2
= 21 + 28i + 7 x (-1)
= 14 + 28i
- Express the given comples number in the form a + ib: (1-i)-(-1+6i)
Solution:
(1 – i )-(-1 + 6i) = 1 – i + 1 – 6i
= 2 – 7i
- Express the given complex number in the form a + ib: (1/5 + i2/5)-(4 + i5/2)
Solution:
(1/5 + i 2/5)-(4 + I 5/2) = 1/5 + 2/5i – 4 – i 5/2
=(1/5 – 4) + i(2/5 – 5/2)
=-19/5 + i(-21/10)
= –19/5 – 21/10 i
- Express the given complex number in the form a + ib:
[(1/3 + i 7/3) + (4 + i 1/3)] – (-4/3 + i)
Solution:
[(1/3 + i 7/3) + (4 + i 1/3)] – (-4/3 + i) = 1/3 + 7/3 i + 4 + 1/3i + 4/3 – i
= ( 1/3 + 4 + 4/3) + i(7/3 + 1/3 -1)
= 17/3 + i 5/3
- Express the given complex number in the form a + ib: (1 – i)4
Solution:
(1 – i)4 = [(1 – i)2]2
= [12 + i2 – 2i] = [1 – 1 – 2i]2 = (-2i)2 = (-2i) x (-2i) = 4i2 = -4
- Express the given complex number in the form a + ib: (1/3 + 3i)2
Solution:
(1/3 + 3i)2 = (1/3)3 + (3i)3 + 3(1/3)(3i)(1/3 + 3i)
= (1/27) + 27i3 + 3(1/3)(3i)(1/3 + 3i)
= 1/27 + 27(-i) + i + 9i2
= 1/27 – 27i + i – 9
= (1/27 – 9) + i(-27 + 1)
= ‑242/27 – 26i
- express he given complex number in the form a + ib: (-2 – 1/3i)3
Solution:
(-2 – 1/3i)3 = (-1)3 (2 + 1/3 i)3
= – [ 23 + (1/3)3 + 3(2)(i/3)(2 + i/3)]
= – [ 8 + (i)3/27 + 2i(2 + i/3)]
= – [ 8 – i/27 + 4i + 2i2/3]
= – [8 – i/27 + 4i – 2/3]
= – [22/3 + 107/27 i ]
= – 22/3 – 107/27 i
- Find the multiplicative inverse of the complex number 4 – 3i
Solution:
Let z = 4 – 3i
Then ź = 4 + 3i and |z|2 = 42 + (-3)2 = 16 + 9 = 25
Therefore the multiplicative inverse of 4 – 3i is given by
z-1 = ź/|z|2 = 4 + 3i/25 = 4/25 + 3/25 i
- Find the multiplicative inverse of the complex number √5 + 3i
Solution:
Let z = √5 + 3i
Then ź = √5 – 3i and |z|2 = (√5)2 + (3i)2 = 5 + 9 = 14
Therefore the multiplicative inverse of √5 + 3i
z-1 = ź/|z|2 = √5 + 3i/14 = √5/14 + 3/14 i
- Find the multiplicative inverse of the complex number –i
Solution:
Let z = -i
Then ź = I and |z|2 = (i)2 = 1
Therefore the multiplicative inverse of –I is given by
z-1 = ź/|z|2 = i/I = i
- Express the following expression in the form of a + ib.
(3 + i√5)(3 – i√5)/(√3 + √2i) – (√3 – √2i)
Solution:
(3 + i√5)(3 – i√5)/(√3 + √2i) – (√3 – √2i) = [32 – (i√5)2] /(√3 + √2i – √3 + √2i )
= (9 – 5i2)/2√2i
= 9+5/2√2i x i/i
= 14i/2√2i2
= 14i/2√2i2
= 14i/2√2(-1)
= -7i/√2 x √2/√2 = -7√2i/2
