161/MR00. problem / Math - Solved Math Problems

MR-161. problem

Determine the value of the expression below:

$\frac{{(1 + i)}^{1000}}{{(1 - i)}^{500}}$

\frac{{(1 + i)}^{1000}}{{(1 - i)}^{500}}

\frac{{(1 + i)}^{1000}}{{(1 - i)}^{500}}

= \frac{{[{(1 + i)}^{2}]}^{500}}{{[{(1 - i)}^{2}]}^{250}}

= \frac{{(1 + 2 i - 1)}^{500}}{{(1 - 2 i - 1)}^{250}}

= \frac{{(2 i)}^{500}}{{(- 2 i)}^{250}}

= \frac{{(2 i)}^{250} \cdot {(2 i)}^{250}}{{(- 2 i)}^{250}}

= {(\frac{2 i}{- 2 i})}^{250} \cdot {(2 i)}^{250}

= {(- 1)}^{250} \cdot {(2 i)}^{250}

= 1 \cdot 2^{250} \cdot i^{250}

= i^{248} \cdot i^{2} \cdot 2^{250}

= 1 \cdot (- 1) \cdot 2^{250}

$= - 2^{250}$

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