Question-1 :- Express the complex number in the form of a + bi:
(5i)(-3i/5)
We have, (5i)(-3i/5) = -15/5 x i2 = -3 x (-1) [i² = -1] = 3
Question-2 :- Express the complex number in the form of a + bi:
i9 + i19
We have, i9 + i19 = (i²)4.i + (i²)9.i = i[(-1)4 + (-1)9] [i² = -1] = i[1 - 1] = 0
Question-3 :- Express the complex number in the form of a + bi:
i-39
We have, i-39 = 1/i39 = 1/(i²)19.i = 1/(-1)19.i [i² = -1] = 1/(-i) = 1/(-i) x (-i)/(-i) = -i/i2 = -i/(-1) = i
Question-4 :- Express the complex number in the form of a + bi:
3(7 + i7) + i (7 + i7)
We have, 3(7 + i7) + i (7 + i7) = 21 + 21i + 7i + 7i2 = 21 + 28i + 7 x (-1) [i² = -1] = 21 - 7 + 28i = 14 + 28i
Question-5 :- Express the complex number in the form of a + bi:
(1 – i) – ( –1 + i6)
We have, (1 – i) – ( –1 + i6) = 1 - i + 1 - 6i = 2 - 7i
Question-6 :- Express the complex number in the form of a + bi:
We have, (1/5 + 2i/5) - (4 + 5i/2) = 1/5 + 2i/5 - 4 - 5i/2 = (1/5 - 4) + (2i/5 - 5i/2) = (1 - 20)/5 + (4i - 25i)/10 = -19/5 + (-21i)/10 = -19/5 - 21i/10
Question-7 :- Express the complex number in the form of a + bi:
We have, [(1/3 + 7i/3) + (4 + i/3)] - (-4/3 + i) = 1/3 + 7i/3 + 4 + i/3 + 4/3 - i = (1/3 + 4/3 + 4) + (7i/3 + i/3 - i) = (1 + 4 + 12)/3 + (7i + i - 3i)/3 = 17/3 + 5i/3
Question-8 :- Express the complex number in the form of a + bi:
(1 - i)4
We have, (1 - i)4 = (1 - i)² x (1 - i)² By using Property, (z₁ + z₂)2 = z₁2 + 2 z₁.z₂ + z₂2 = [1² + i² + 2i] x [1² + i² + 2i] [i² = -1] = [1 - 1 + 2i] x [1 - 1 + 2i] = 2i x 2i = 4i² = 4 x (-1) [i² = -1] = -4
Question-9 :- Express the complex number in the form of a + bi:
(1/3 + 3i)³
We have, (1/3 + 3i)3 By using Property, (z₁ + z₂)3 = z₁3 + z₂3 + 3 z₁2.z₂ + 3 z₁.z₂2 = (1/3)³ + (3i)³ + 3 x (1/3)² x 3i + 3 x 1/3 x (3i)² = 1/27 + 27i³ + 9i x 1/9 + 9i² = 1/27 + 27 x (-i) + i + 9 x (-1) [i² = -1; i³ = -i] = 1/27 - 27i + i - 9 = (1/27 - 9) - 26i = -242/27 - 26i
Question-10 :- Express the complex number in the form of a + bi:
(-2 - i/3)³
We have, (-2 - i/3)³ By using Property, (z₁ + z₂)3 = z₁3 + z₂3 + 3 z₁2.z₂ + 3 z₁.z₂2 = (-2)³ + (-i/3)³ + 3 x (-2)² x (-i/3) + 3 x (-2) x (-i/3)² = -8 + (-i)³/27 - 4i - 6 x i²/9 = -8 + i/27 - 4i + 2/3 [i² = -1; (-i)³ = i] = (-8 + 2/3) + (i/27 - 4i) = -22/3 - 107i/27
Question-11 :- Find the multiplicative inverse of the complex numbers:
4 - 3i
Let z = 4 – 3i
Conjugate z = 4 + 3i |z|2 = 42 + (-3)2 = 16 + 9 = 25 Therefore, the multiplicative inverse of 4 - 3i is given by z-1 = z/|z|2 z-1 = (4 + 3i)/25 = 4/25 + 3i/25
Question-12 :- Find the multiplicative inverse of the complex numbers:
√5 + 3i
Let z = √5 + 3i
Conjugate z = √5 - 3i |z|2 = √52 + 32 = 5 + 9 = 14 Therefore, the multiplicative inverse of √5 + 3i is given by z-1 = z/|z|2 z-1 = (√5 - 3i)/14 = √5/14 - 3i/14
Question-13 :- Find the multiplicative inverse of the complex numbers:
-i
Let z = 0 – 1.i
Conjugate z = 0 + 1.i |z|2 = 0 + (-1)2 = 0 + 1 = 1 Therefore, the multiplicative inverse of -i is given by z-1 = z/|z|2 z-1 = (0 + i)/1 = i
Question-14 :- Express the following expression in the form of a + ib :