Question

Se considera  numerele reale    a=$\sqrt{3+2 \sqrt{2} }$ si b=$\sqrt{3-2 \sqrt{2}}$.
 a) Aratati ca $a^{2} -2ab + b^{2} = 4$
b) Demonstrati ca $\frac{1}{a} - \frac{1}{b}$ ∈ Q

Answer 1

[√(3+2√2) - √(3-2√2) ]² =
=3+2√2 -2(√9-8)+3-2√2=
=3 -2*1+3=4

1/√(3+2√2)-1/√(3-2√2) =
=[√(3-2√2)-√(3+2√2)]/√(3+2√2)√(3-2√2) =
=[√(3-2√2)-√(3+2√2)]/√(9-8) =
=√(3-2√2)-√(3+2√2)