Simplify : $\sqrt{\frac{a}{2}}+\sqrt{\frac{a}{2}}\ge \sqrt{a}$? Just out of curiosity, is $$\sqrt{\frac{a}{2}}+\sqrt{\frac{a}{2}}\ge \sqrt{a},a \gt0\quad?$$  
Thanks

$$\sqrt{\frac{a}{2}} + \sqrt{\frac a2} = 2 \frac{\sqrt a}{\sqrt 2} = \frac {2}{\sqrt 2} \sqrt{a} = \sqrt 2 \sqrt a = \sqrt{2a} > \sqrt a$$

Added: perhaps your question originates from not remembering that $$(a + b)^2 \not\equiv a^2 + b^2\;?$$ If we do square each side of the proposed inequality (which we can do, without having to worry about whether doing so might change the direction of the inequality, since $a\gt 0$), then note: $$\left(\sqrt{\frac a2} + \sqrt{\frac a2}\right)^2\;\;  \geq \; \;(\sqrt a)^2$$ $$ \iff \frac a2 + 2\cdot \frac a2 + \frac a2 \;\;\geq \;\; a $$ $$\iff 2a \geq a\quad \checkmark$$
