Question

aratati ca numarul a este natural, unde:
a.) a=
$\sqrt{30 - 12 \sqrt{6} } + 2 \sqrt{4 - 2 \sqrt{3} } - 3 \sqrt{3 - 2 \sqrt{2} }$
b.) a=
$( \sqrt{8 - 2 \sqrt{15} } + \sqrt{8 + 2 \sqrt{15} } )^{2}$
​

Answer 1

Explicație pas cu pas:

a)

$\sqrt{30 - 12 \sqrt{6} } + 2 \sqrt{4 - 2 \sqrt{3} } - 3 \sqrt{3 - 2 \sqrt{2} } =$

$= \sqrt{18 - 12 \sqrt{6} + 12} + 2 \sqrt{3 - 2 \sqrt{3} + 1} - 3 \sqrt{2 - 2 \sqrt{2} + 1}$

$= \sqrt{ {(3 \sqrt{2} )}^{2} - 2 \cdot 3 \sqrt{2} \cdot 2 \sqrt{3} + {(2 \sqrt{3} )}^{2} } + 2 \sqrt{ {( \sqrt{3} )}^{2} - 2 \cdot \sqrt{3} \cdot 1 + {1}^{2} } - 3 \sqrt{ {( \sqrt{2} )}^{2} - 2 \sqrt{2} \cdot 1 + {1}^{2} }$

$= \sqrt{ {(3 \sqrt{2} - 2 \sqrt{3} )}^{2}} + 2 \sqrt{ {( \sqrt{3} - 1)}^{2}} - 3 \sqrt{ {( \sqrt{2} - 1)}^{2}}$

$= |3 \sqrt{2} - 2 \sqrt{3}| + 2 |\sqrt{3} - 1| - 3 |\sqrt{2} - 1|$

$= 3 \sqrt{2} - 2 \sqrt{3} + 2 \sqrt{3} - 2 - 3 \sqrt{2} + 3$

$= \bf 1$

b)

$( \sqrt{8 - 2 \sqrt{15} } + \sqrt{8 + 2 \sqrt{15} } )^{2} =$

$= 8 - 2 \sqrt{15} + 2 \sqrt{(8 - 2 \sqrt{15})(8 + 2 \sqrt{15})} + 8 + 2 \sqrt{15}$

$= 16 + 2 \sqrt{ {8}^{2} - {(2 \sqrt{15})}^{2} } = 8 - \sqrt{64 - 60}$

$= 8 - \sqrt{4} = 8 - 2 = \bf 6$

$q.e.d.$