Obie strony równości podnosimy do kwadratu.

${\left(\sqrt{3-2\sqrt{2}}+\sqrt{18-8\sqrt{2}}\right)}^2=9$  

Mamy więc:

${\left(\sqrt{3-2\sqrt{2}}+\sqrt{18-8\sqrt{2}}\right)}^2=$  

$={\left(\sqrt{3-2\sqrt{2}}\right)}^2+2\cdot \sqrt{3-2\sqrt{2}}\cdot \sqrt{18-8\sqrt{2}}+{\left(\sqrt{18-8\sqrt{2}}\right)}^2=$  

$=3-2\sqrt{2}+2\cdot \sqrt{\left(3-2\sqrt{2}\right)\left(18-8\sqrt{2}\right)}+18-8\sqrt{2}=$  

$=21-10\sqrt{2}+2\cdot \sqrt{54-24\sqrt{2}-36\sqrt{2}+16\cdot 2}=$  

$=21-10\sqrt{2}+2\cdot \sqrt{54-60\sqrt{2}+32}=21-10\sqrt{2}+2\cdot \sqrt{86-60\sqrt{2}}=$  

$=21-10\sqrt{2}+2\cdot \sqrt{50-60\sqrt{2}+36}=$  

$=21-10\sqrt{2}+2\cdot \sqrt{{\left(5\sqrt{2}\right)}^2-2\cdot 5\sqrt{2}\cdot 6+6^2}=$  

$=21-10\sqrt{2}+2\cdot \sqrt{{\left(5\sqrt{2}-6\right)}^2}=21-10\sqrt{2}+2\cdot \left(5\sqrt{2}-6\right)=$  

$=21-10\sqrt{2}+10\sqrt{2}-12=9$  

Równość jest prawdziwa.