Complex Power/Examples/(2+i)^4

Example of Complex Power

$\paren {2 + i}^4 = -7 + 24 i$

We have that:

$\paren {2 + i}^4 = \paren {\paren {2 + i}^2}^2$

Hence:

$\ds \paren {2 + i} \paren {2 + i}$

$=$

$\ds \paren {2 \times 2 - 1 \times 1} + \paren {2 \times 1 + 1 \times 2} i$

$\ds $

$=$

$\ds \paren {4 - 1} + \paren {2 + 2} i$

$\ds $

$=$

$\ds 3 + 4 i$

Then:

$\ds \paren {3 + 4 i} \paren {3 + 4 i}$

$=$

$\ds \paren {3 \times 3 - 4 \times 4} + \paren {3 \times 4 + 4 \times 3} i$

$\ds $

$=$

$\ds \paren {9 - 16} + \paren {12 + 12} i$

$\ds $

$=$

$\ds -7 + 24 i$

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $1 \ \text{(iii)}$