Question

a = ( 2+ √3 ) la a 2-a - √2 ( √5+ √3 ) - 2√3 ( 2 - √2 ) + √2 ( √5 - √3 ) , aratati ca a E ℕ .
REPEDE VA ROGGGGGGG

Answer 1

A = ( 2+ √3 )² - √2 ( √5+ √3 ) - 2√3 ( 2 - √2 ) + √2 ( √5 - √3 ) =
= (4+ 4√3+3 ) - (√10+ √6 ) - (4√3  - 2√6 ) +  ( √10 - √6 ) =
=7+ 4√3 - √10 - √6  - 4√3 + 2√6  +   √10 - √6 = 
=7     7∈ N

Answer 2

$A=(2+ \sqrt{3} )^2- \sqrt{2} ( \sqrt{5} + \sqrt{3} )-2 \sqrt{3} (2- \sqrt{2} )+ \sqrt{2} ( \sqrt{5} - \sqrt{3} ) \\ A=2^2+2 \cdot 2 \cdot \sqrt{3} + \sqrt{3}^2 - \sqrt{10} - \sqrt{6} -4 \sqrt{3} +2 \sqrt{6} + \sqrt{10} - \sqrt{6} \\ A=4+4 \sqrt{3} + 3 - \sqrt{10} - \sqrt{6} -4 \sqrt{3} +2 \sqrt{6} + \sqrt{10} - \sqrt{6} \\ A=4+3 \\ A=7 \in N$