Answer
4
Work Step by Step
$\sqrt (13 + \sqrt 2 + \frac{7}{3 + \sqrt 2})$ ........(1)
To simplify first rationalize $\frac{7}{3 + \sqrt 2} $
To rationalize the denominator multiply the numerator and denominator by the conjugate of $3 + \sqrt 2$.
The conjugate of $3 + \sqrt 2$ is $3 - \sqrt 2$
$\frac{7}{3 + \sqrt 2} \times \frac{3 - \sqrt 2}{3 - \sqrt 2}$
$= \frac{7(3 - \sqrt 2)}{(3 + \sqrt 2)(3 - \sqrt 2)}$
$( \sqrt a - \sqrt b)( \sqrt a + \sqrt b)$ = $ (\sqrt a)^{2}$ - $ (\sqrt b)^{2}$. Therefore,
$( 3 + \sqrt 2)( 3 - \sqrt 2)$ = $ (3^{2})-(\sqrt 2^{2})$.
$= \frac{7(3 - \sqrt 2)}{(3^{2})-(\sqrt 2^{2})}$
$= \frac{7(3 - \sqrt 2)}{9-2}$
$= \frac{7(3 - \sqrt 2)}{7}$
$=3 - \sqrt 2$
Substitute this value in Equation (1)
$\sqrt (13 + \sqrt 2 + 3 - \sqrt 2)$
$\sqrt (13 + 3 + \sqrt 2 - \sqrt 2)$
$\sqrt 16$ $ = 4 $
$ = 4 $
