Question

Se consideră numerele a=
$2 \sqrt{3} + 3$
si b=
$\sqrt{21 - 12 \sqrt{3} }$
Media geometrica a numerelor a si b este:
A. 3
B. 2√3
C. 4√3
D. √3​

Answer 1

$\begin{array}{lcl}a &=& 2\sqrt{3}+3\end{array} \\\\ \begin{array}{lcl} b &=& \sqrt{21-12\sqrt 3} \\ &=& \sqrt{3\cdot (7-4\sqrt 3)} \\ &=&\sqrt{3}\cdot \sqrt{(2-\sqrt 3)^2} \\&=& \sqrt{3}\cdot |2-\sqrt{3}|\\&=&\sqrt{3}\cdot (2-\sqrt 3) \\ &=&2\sqrt 3 - 3 \end{array}\\ \\ \\ \begin{array}{lcl}\sqrt{ab}&=&\sqrt{(2\sqrt 3 + 3)(2\sqrt 3 - 3)} \\\\ &=&\sqrt{(2\sqrt 3)^2 - 3^2} \\\\&=& \sqrt{4\cdot 3 - 9} \\ \\&=& \boxed{\sqrt{3}}\\ \\\end{array}$

Răspuns:  D)