a)
\sqrt{13-4\sqrt3}=2\sqrt3-1 \ |^2 obustronnie do kwadratu
\sqrt{(13-4\sqrt3)^2}=(2\sqrt3-1)^2
13-4\sqrt3=4*3-4\sqrt3+1 \ |+4\sqrt3
13=12+1
13=13
L=P
b)
\sqrt{9-4\sqrt5}-\sqrt{6-2\sqrt5}=-1 \ |^2
9-4\sqrt5-2\sqrt{(9-4\sqrt5)(6-2\sqrt5)}+6-2\sqrt5=1
9-4\sqrt5-2\sqrt{54-18\sqrt5-24\sqrt5+8*5}+6-2\sqrt5=1
9-4\sqrt5-2\sqrt{94-42\sqrt5}+6-2\sqrt5=1
15-6\sqrt5-2\sqrt{94-42\sqrt5}=1
15-6\sqrt5-2\sqrt{49-42\sqrt5+45}=1
{15-6\sqrt2-2\sqrt{(7-3\sqrt5)^2}=1 ................(7-3\sqrt5)^2=49-2*7*3\sqrt5+9*5=94-42\sqrt5}
15-6\sqrt2-2(7-3\sqrt5)=1
15-6\sqrt2-14+6\sqrt5=1
1=1
L=P
c)
\sqrt[4]{17-12\sqrt2}*(3+2\sqrt2)^{\frac{1}{2}}=1 \ |^4 obustronnie do potęgi czwartej
({17-12\sqrt2})*(3+2\sqrt2)^{2}=1
{L=({17-12\sqrt2})*(3+2\sqrt2)^{2}=({17-12\sqrt2})*(9+12\sqrt2+4*2)=(17-12\sqrt2)(17+12\sqrt2)=}
=17^2-(12\sqrt2)^2=289-144*2=289-288=1
1=1
L=P
d)
\sqrt{5-2\sqrt6}*(49+20\sqrt6)^{\frac{1}{4}}=1 \ |^4
((5-2\sqrt6)^{\frac{1}{2}})^4*(49+20\sqrt6)=1
{L=((5-2\sqrt6)^{\frac{1}{2}})^4*(49+20\sqrt6)=(5-2\sqrt6)^2*(49+20\sqrt6)=(25-20\sqrt6+4*6)(49+20\sqrt6)=}
=(49-20\sqrt6)(49+20\sqrt6)=49^2-(20\sqrt6)^2=2401-400*6=2401-2400=1
1=1
L=P