Express the complex numbers in the standard form a + ib: (1 + 2i)-3

(1 + 2i)^-3

Let us simplify and express in the standard form of (a + ib),

(1 + 2i)^-3 = \(\dfrac{ 1}{(1 + 2i)^3}\)

= \( \dfrac{1}{(1+6i+4i^2+8i^3)}\)

=\( \dfrac{1}{(1+6i+4(-1)+8i^2.i)}\) [since, i² = -1]

= \( \dfrac{-1}{(3+2i)}\)

[Multiply and divide with (3-2i)]

=\( \dfrac{-1}{(3+2i)}× \dfrac{(3-2i)}{(3-2i)}\)

= \(\dfrac{ (-3+2i) }{ (9-4(-1))}\)

=\(\dfrac{ (-3+2i) }{13}\)

The values of a, b are \(\dfrac{ -3}{13}, \dfrac{2}{13}\)