Express the following complex numbers in the standard form a + ib: (i) $(1 + 2i)^{-3}$ (ii) (3 – 4i) / [(4

Home » Express the following complex numbers in the standard form a + ib: (i) (ii) (3 – 4i) / [(4 – 2i) (1 + i)]

Express the following complex numbers in the standard form a + ib:
(i) $(1 + 2i)^{-3}$
(ii) (3 – 4i) / [(4 – 2i) (1 + i)]

Express the following complex numbers in the standard form a + ib:
(i) $(1 + 2i)^{-3}$
(ii) (3 – 4i) / [(4 – 2i) (1 + i)]

Solution:

(i) $(1+2 \mathrm{i})^{-3}$
Simplify and express in the standard form of $(a+i b)$ ,
$\begin{array}{l} (1+2 i)^{-3}=1 /(1+2 i)^{3} \\ =1 /\left(1^{3}+3(1)^{2}(2 i)+2(1)(2 i)^{2}+(2 i)^{3}\right) \\ =1 /\left(1+6 i+4 i^{2}+8 i^{3}\right) \\ =1 /\left(1+6 i+4(-1)+8 i^{2} . i\right)\left[\text { since, } i^{2}=-1\right] \\ =1 /(-3+6 i+8(-1) i)\left[\text { since, } i^{2}=-1\right] \\ =1 /(-3-2 i) \\ =-1 /(3+2 i) \end{array}$
Multiplying and dividing with $(3-2i)$
$\begin{array}{l} =-1 /(3+2 i) \times(3-2 i) /(3-2 i) \\ =(-3+2 i) /\left(3^{2}-(2 i)^{2}\right) \\ =(-3+2 i) /\left(9-4 i^{2}\right) \\ =(-3+2 i) /(9-4(-1)) \\ =(-3+2 i) / 13 \end{array}$
As a result, the values of $\mathrm{a}, \mathrm{b}$ are $-3 / 13,2 / 13$

(ii) $(3-4 i) /[(4-2 i)(1+i)]$
Simplify and express in the standard form of $(a+i b)$ ,
$\begin{array}{l} (3-4 i) /[(4-2 i)(1+i)]=(3-4 i) /[4(1+i)-2 i(1+i)] \\ =(3-4 i) /\left[4+4 i-2 i-2 i^{2}\right] \\ =(3-4 i) /[4+2 i-2(-1)]\left[\text { since, } i^{2}=-1\right] \\ =(3-4 i) /(6+2 i) \end{array}$
Multiplying and dividing with $(6-2i)$
$\begin{array}{l} =(3-4 i) /(6+2 i) \times(6-2 i) /(6-2 i) \\ =[3(6-2 i)-4 i(6-2 i)] /\left(6^{2}-(2 i)^{2}\right) \\ =\left[18-6 i-24 i+8 i^{2}\right] /\left(36-4 i^{2}\right) \\ =[18-30 i+8(-1)] /(36-4(-1))\left[\text { since, } i^{2}=-1\right] \\ =[10-30 i] / 40 \\ =(1-3 i) / 4 \end{array}$
As a result, the values of $\mathrm{a}, \mathrm{b}$ are $1 / 4,-3 / 4$