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Express the following in the form $\;a+ib : \large\frac{5 + \sqrt{2} i }{1 - \sqrt{2} i}$

$(a)\;1+\sqrt{2} i\qquad(b)\;0\qquad(c)\;1- 2 \sqrt{2} i\qquad(d)\;1+ 2 \sqrt{2} i$

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Answer : $\;1+2\sqrt{2} i$
Explanation :
$\large\frac{5 + \sqrt{2} i }{1 - \sqrt{2} i}= \large\frac{5 + \sqrt{2} i }{1 - \sqrt{2} i} \times \large\frac{1+\sqrt{2} i}{1+\sqrt{2} i}$
$ = \large\frac{5 + 5 \sqrt{2} i + \sqrt{2} i + 2 i^{2}}{1^{2} - (\sqrt{2}i)^{2}}$
$ = \large\frac{3 + 6 \sqrt{2} i}{1+2}$
$= \large\frac{3 (1 + 2 \sqrt{2} i)}{3}$
$= 1+2\sqrt{2} i$
answered Apr 16, 2014 by yamini.v
 

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