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Complex Numbers — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers2 Marks
Question
Express the following complex numbers in the standard form a + ib:
$(1-2\text{i})^{-3}$
✓

Answer

$(1-2\text{i})^{-3}=\frac{1}{(1+2\text{i}^3)} \ \Big(\therefore \ \text{z}^{-3}=\frac{1}{\text{z}^3}\Big)$
$=\frac{1}{1^3+(2\text{i})^3+3\times1\times2\text{i}(1+2\text{i})}$
$=\frac{1}{1^3+2^3\times\text{i}^3+6\text{i}(1+2\text{i})}$
$=\frac{1}{1-8\text{i}+6\text{i}-12} \ \big(\therefore \ \text{i}^3=-\text{i} \ \text{and} \ \text{i}^2=-1\big)$
$=\frac{1}{-11-2\text{i}}$
$=\frac{1}{-11-2\text{i}}\times\frac{(-11+2\text{i})}{(-11+2\text{i})}$
$=\frac{-11+2\text{i}}{121+4}$
$=\frac{-11}{125}+\frac{2}{125}\text{i}$
$\therefore \ (1-2\text{i})^{-3}=\frac{-11}{125}+\frac{2}{125}\text{i}$

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