Chapter 5 - Number Theory and the Real Number System - 5.4 The Irrational Numbers - Exercise Set 5.4 - Page 299: 106

$\dfrac{3\sqrt2}{2}$

Rationalize the denominator to obtain: \begin{align*} \sqrt2+\sqrt{\frac{1}{2}}&=\sqrt2 + \sqrt{\frac{1}{2} \cdot \frac{2}{2}}\\ &=\sqrt2 +\sqrt{\frac{2}{4}}\\ &=\sqrt2 + \sqrt{\frac{1}{4} \cdot 2}\\ &=\sqrt2+\frac{1}{2}\sqrt2\\ &=\sqrt2\left(1+\frac{1}{2}\right)\\ &=\sqrt2\left(\frac{3}{2}\right)\\ &=\frac{3\sqrt2}{2} \end{align*}