Question

Express the following complex numbers in the form of a+ib with explanation
1.(2+i)(4+i)
2.3+4i÷2+3i
3.1+i÷2+4i
4.(2+3i)(3+5i)
5.1+i÷2-3i​

Answer 1

Step-by-step explanation:

(i) [(1 + i) (1 + √3i)]/(1 – i) Now let us simplify and express in the standard form of (a + ib), [(1 + i) (1 + √3i)]/(1 – i) = [1(1 + √3i) + i(1 + √3i)]/(1 - i) = (1 + √3i + i + √3i2)/(1 – i) = (1 + (√3 + 1)i + √3(-1))/(1 - i) [since, i2 = -1] = [(1 - √3) + (1 + √3)i]/(1 - i) [by multiply and divide with (1 + i)] = [(1 - √3) + (1 + √3)i]/(1 - i) × (1 + i)/(1 + i) = [(1 - √3) (1 + i) + (1 + √3)i(1 + i)]/(12 – i2) = [1 - √3 + (1 - √3)i + (1 + √3)i + (1 + √3)i2]/(1 - (-1)) [since, i2 = -1] = [(1 - √3) + (1 - √3 + 1 + √3)i + (1+ √3)(-1)]/2 = (-2√3 + 2i)/2 = -√3 + i Thus the values of a, b are -√3, i (ii) (2 + 3i)/(4 + 5i) Now let us simplify and express in the standard form of (a + ib), (2 + 3i)/(4 + 5i) = [multiply and divide with (4 - 5i)] = (2 + 3i)/(4 + 5i) × (4 - 5i)/(4 - 5i) = [2(4 - 5i) + 3i(4 - 5i)]/(42 – (5i)2) = [8 – 10i + 12i – 15i2]/(16 – 25i2) = [8 + 2i - 15(-1)]/(16 – 25(-1)) [since, i2 = -1] = (23 + 2i)/41 ∴ The values of a, b are 23/41, 2i/41 (iii) (1 – i)3/(1 – i3) Now let us simplify and express in the standard form of (a + ib), (1 – i)3/(1 – i3) = [13 – 3(1)2i + 3(1)(i)2 – i3]/(1 - i2.i) = [1 – 3i + 3(-1)-i2.i]/(1 – (-1)i) [since, i2 = -1] = [-2 – 3i – (-1)i]/(1 + i) = [-2 - 4i]/(1 + i) [By Multiply and divide with (1 - i)] = [-2 - 4i]/(1 + i) × (1 - i)/(1 - i) = [-2(1 - i) -4i(1 - i)]/(12 – i2) = [-2 + 2i - 4i + 4i2]/(1 – (-1)) = [-2 - 2i + 4(-1)]/2 = (-6 - 2i)/2 = -3 – i ∴ The values of a, b are -3, -i (iv) (1 + 2i)-3 Now let us simplify and express in the standard form of (a + ib), (1 + 2i)-3 = 1/(1 + 2i)3 = 1/(13 + 3(1)2 (2i) + 2(1)(2i)2 + (2i)3) = 1/(1 + 6i + 4i2 + 8i3) = 1/(1 + 6i + 4(-1) + 8i2.i) [since, i2 = -1] = 1/(-3 + 6i + 8(-1)i) [since, i2 = -1] = 1/(-3 - 2i) = -1/(3 + 2i) [By multiply and divide with (3 - 2i)] = -1/(3 + 2i) × (3 - 2i)/(3 - 2i) = (-3 + 2i)/(32 – (2i)2) = (-3 + 2i)/(9-4i2) = (-3 + 2i)/(9 - 4(-1)) = (-3 + 2i)/13 ∴ The values of a, b are -3/13, 2i/13 (v) (3 – 4i)/[(4 – 2i) (1 + i)] Now let us simplify and express in the standard form of (a + ib), (3 – 4i)/[(4 – 2i) (1 + i)] = (3 - 4i)/[4(1 + i) - 2i(1 + i)] = (3 - 4i)/[4 + 4i - 2i - 2i2] = (3 - 4i)/[4 + 2i - 2(-1)] [since, i2 = -1] = (3 - 4i)/(6 + 2i) [By multiply and divide with (6 - 2i)] = (3 - 4i)/(6 + 2i) × (6 - 2i)/(6 - 2i) = [3(6 - 2i) - 4i(6 - 2i)]/(62 – (2i)2) = [18 – 6i – 24i + 8i2]/(36 – 4i2) = [18 – 30i + 8 (-1)]/(36 – 4 (-1)) [since, i2 = -1] = [10 - 30i]/ 40 = (1 – 3i)/4 Thus the values of a, b are 1/4, -3i/4Read more on Sarthaks.com - https://www.sarthaks.com/659743/express-the-following-complex-numbers-in-the-standard-form-a-ib-i-1-i-3i-1-i-ii-2-3i-4-5i-iii-1-i3