A first course in Complex Analysis with Applications Assume for the moment that $\sqrt{1+i}$. makes sense in the complex number system. 
How would you then demonstrate the validity of the equality
$$\sqrt{1+i} = \sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{2}} + i\sqrt{-\frac{1}{2} + \frac{1}{2} \sqrt{2}}$$?
Source: Dennis G. Zill & Patrick D. Shanahan
I'm Confused that what to do to solve this one or what the question asks for?

In order to prove  o$$\sqrt{1+i} = \sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{2}} + i\sqrt{-\frac{1}{2} + \frac{1}{2} \sqrt{2}}$$
we square the right hand side and show that it is the same as $1+i.$
Note that $$\bigg (\sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{2}} + i\sqrt{-\frac{1}{2} + \frac{1}{2} \sqrt{2}}\bigg )^2 =$$
$$ \bigg( \frac{1}{2} +\frac{1}{2} \sqrt{2}\bigg) -\bigg( \frac{-1}{2} +\frac{1}{2} \sqrt{2}\bigg) + 2i  \sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{2}}    \sqrt{-\frac{1}{2} + \frac{1}{2} \sqrt{2}}=$$
$$ 1 +2i(1/2)=1+i$$
