Question

solve the above question.....

Answer 1

$Given , (5 + 2 \sqrt{6} )^2 + \frac{1}{(5 + 2 \sqrt{6})^2}$

We know that (a + b)^2 = a^2 + b^2 + 2ab.

$(5)^2 + (2 \sqrt{6})^2 + 2(5)(2 \sqrt{6}) + \frac{1}{(5)^2 + (2 \sqrt{6})^2 + 2(5)(2 \sqrt{6}) }$

$25 + 4 * 6 + 20 \sqrt{6} + \frac{1}{25 + 4 * 6 + 20 \sqrt{6} }$

$25 + 24 + 20 \sqrt{6} + \frac{1}{25 + 24 + 20 \sqrt{6} }$

$49 + 20 \sqrt{6} + \frac{1}{49 + 20 \sqrt{6} }$

On rationalizing, we get

$49 + 20 \sqrt{6} + \frac{1}{49 + 20 \sqrt{6} } * \frac{49 - 20 \sqrt{6} }{49 - 20 \sqrt{6}}$

We know that (a + b)(a - b) = a^2 - b^2

$49 + 20 \sqrt{6} + \frac{1 * 49 - 20 \sqrt{6}}{(49)^2 - (20 \sqrt{6})^2 }$

$49 + 20 \sqrt{6} + \frac{49 - 20 \sqrt{6} }{2401 - 2400}$

$49 + 20 \sqrt{6} + \frac{49 - 20 \sqrt{6} }{1}$

$49 + 20 \sqrt{6} + 49 - 20 \sqrt{6}$

49 + 49

98.


Hope this helps!