Dana jest liczba:

${\left(0,5^{\sqrt{2}}\cdot 2\right)}^{\sqrt{2}+1}$

Zauważmy, że:

$0,5=\frac{1}{2}=2^{-1}$

Zatem możemy zapisać:

${\left(0,5^{\sqrt{2}}\cdot 2\right)}^{\sqrt{2}+1}={\left({\left(2^{-1}\right)}^{\sqrt{2}}\cdot 2\right)}^{\sqrt{2}+1}=$

$={\left(2^{-1\cdot \sqrt{2}}\cdot 2\right)}^{\sqrt{2}+1}={\left(2^{-\sqrt{2}}\cdot 2^1\right)}^{\sqrt{2}+1}=$

$={\left(2^{-\sqrt{2}+1}\right)}^{\sqrt{2}+1}={\left(2^{1-\sqrt{2}}\right)}^{1+\sqrt{2}}=$

$=2^{\left(1-\sqrt{2}\right)\left(1+\sqrt{2}\right)}=2^{1^2-{\left(\sqrt{2}\right)}^2}=$

$=2^{1-2}=2^{-1}=\frac{1}{2}$

Odp. A