Question

what is the value of this frac​

Answer 1

$\blue{\bf{\dfrac{\sqrt{3} + 1}{3}}}$

Step-by-step explanation:

To Solve:

$\dfrac{2 \sqrt{2} + \sqrt{6} }{3 \sqrt{2 + \sqrt{3} } }$

Solution:

$\dfrac{2 \sqrt{2} + \sqrt{6} }{3 \sqrt{2 + \sqrt{3} } } \\ \\ \frac{2 \sqrt{2} + 2 \sqrt{2}. \sqrt{3}}{3 \sqrt{2 + \sqrt{3}}} \\ \\ \frac{ \sqrt{2}(2 + \sqrt{3})}{3 \sqrt{2 + \sqrt{3}}} \\ \\ \frac{ \sqrt{2}( \sqrt{2 + \sqrt{3}})( \sqrt{2 + \sqrt{3}}) }{3( (\sqrt{2 + \sqrt{3}})} \\ \\ \frac{ \sqrt{2}( \sqrt{2 + \sqrt{3}}) \cancel{( \sqrt{2 + \sqrt{3}})} }{3\cancel{(\sqrt{2 + \sqrt{3}}})} \\ \\ \frac{ \sqrt{2}(\sqrt{2 + \sqrt{3} })}{3} \\ \\ \frac{ \sqrt{2 \times 2 + 2 \times \sqrt{3}}}{3} \\ \\ \frac{ \sqrt{4 + 2 \sqrt{3}}}{3} \\ \\ \frac{ \sqrt{3 + 1 + 2 \sqrt{3}} }{3} \\ \\ \frac{ \sqrt{( \sqrt{3} + 1) {}^{2} } }{3} \\ \\ \frac{ \sqrt{3} + 1}{3}$

Therefore, the required answer is $\pink{\bf{\dfrac{\sqrt{3} + 1}{3}}}$