Express the following complex number in polar form and exponential form: $\frac{(1 + 7 i)}{(2 - i)^{2}}$

\sqrt{2} (- c o s \frac{3 π}{4} + i s i n \frac{3 π}{4}); - \sqrt{2} e^{i ⎛ ⎝ \frac{3 π}{4} ⎞ ⎠}

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\sqrt{2} (c o s \frac{3 π}{4} - i s i n \frac{3 π}{4}); \sqrt{2} e^{i ⎛ ⎝ \frac{- 3 π}{4} ⎞ ⎠}

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\sqrt{2} (c o s \frac{3 π}{4} + i s i n \frac{3 π}{4}); - \sqrt{2} e^{i ⎛ ⎝ \frac{3 π}{4} ⎞ ⎠}

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\sqrt{2} (c o s \frac{3 π}{4} + i s i n \frac{3 π}{4}); \sqrt{2} e^{i ⎛ ⎝ \frac{- 3 π}{4} ⎞ ⎠}

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Solution

The correct option is C $\sqrt{2} (c o s \frac{3 π}{4} + i s i n \frac{3 π}{4}); - \sqrt{2} e^{i ⎛ ⎝ \frac{3 π}{4} ⎞ ⎠}$
Simplifying, we get
$\frac{(1 + 7 i) (2 + i)^{2}}{25}$
$= \frac{(1 + 7 i) (3 + 4 i)}{25}$
$= \frac{- 25 + 25 i}{25}$
$= - 1 + i$
$= \sqrt{2} . (\frac{- 1 + i}{\sqrt{2}})$
$= \sqrt{2} . e^{i . \frac{3 π}{4}}$

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z = \frac{(1 + 7 i)}{(2 - i)^{2}}

The polar form of complex number 2 + 2i

- 3 \sqrt{2} + 3 \sqrt{2} i

The polar form of complex number 1 - i

{⎛ ⎝ \sqrt{{(\sqrt{2} - \frac{3}{2})}^{2}} - \sqrt[3]{(1 - \sqrt{2})^{3}} ⎞ ⎠}^{2} + 2^{- \frac{3}{2}} cos \frac{3 π}{4}

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