Prove that


Express $(1-2 i)^{-3}$ in the form $(a+i b)$


We have, $(1-2 i)^{-3}$

$\Rightarrow \frac{1}{(1-2 i)^{3}}=\frac{1}{1-8 i^{3}-6 i+12 i^{2}}=\frac{1}{1+8 i-6 i-12}=\frac{1}{2 i-11}$

$\Rightarrow \frac{1}{-11+2 i}$

$=\frac{1}{-11+2 i} \times \frac{-11-2 i}{-11-2 i}$

$=\frac{-11-2 i}{(-11)^{2}-(2 i)^{2}}=\frac{-11-2 i}{121+4}$

$=\frac{-11-2 i}{125}$

$=\frac{-11}{125}-\frac{2 i}{125}$


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