Evaluate-3-2-

Question Number 17713 by tawa tawa last updated on 09/Jul/17

Evaluate : {(- \sqrt{3})}^{(- \sqrt{2})}

Commented by b.e.h.i.8.3.417@gmail.com last updated on 09/Jul/17

a = {(- \sqrt{3})}^{(- \sqrt{2})}

l n a = - \sqrt{2} l n (- \sqrt{3}) = - \sqrt{2} l n (i^{2} \sqrt{3}) =

= - \sqrt{2} [2 l n i + \frac{1}{2} l n 3] = - \sqrt{2} [2 i \frac{π}{2} + \frac{1}{2} l n 3] =

= - \frac{\sqrt{2}}{2} (2 i π + l n 3)

\Rightarrow a = e^{- \frac{\sqrt{2}}{2} (2 i π + l n 3)} = {(e^{2 i π + l n 3})}^{\frac{- \sqrt{2}}{2}} =

= {[{(e^{i π})}^{2} . e^{l n 3}]}^{- \frac{\sqrt{2}}{2}} = {[{(- 1)}^{2} . 3]}^{- \frac{\sqrt{2}}{2}} = 3^{(- \frac{\sqrt{2}}{2})}

n o t e :

1) i = c o s 90 + i s i n 90 = e^{i \frac{π}{2}} \Rightarrow l n i = i . \frac{π}{2}

2) e^{i π} + 1 = 0 ({E u l e r}^{'} s f o r m u l a)

Commented by tawa tawa last updated on 09/Jul/17

God bless you sir .