<s>[INST] Simplifying Algebraic Expression Under A Square Root This is part of a problem for calculating the length of a curve. Unfortunately, I'm stuck on a pretty basic algebra concept. 
Solutions for my problem say that:
$\sqrt{\frac{1}{2t} + 1 + \frac{t}{2}} = \frac{\sqrt{t} + \frac{1}{\sqrt{t}}}{\sqrt{2}}$ 
I cannot understand how they got this at all. 
The most I can simplify the original expression is as 
$\sqrt{\frac{1}{2}(\frac{1}{t} + 2 + t)} = \sqrt{1/2}\sqrt{\frac{1}{t} + 2 + t}  = \frac{1}{\sqrt{2}}\sqrt{\frac{1}{t} + 2 + t}$
I can't see how they get simplify the algebraic expression under the square root.
 [/INST] Put over a common denominator: $$\sqrt{\frac{1}{2t} + 1 + \frac{t}{2}} =\sqrt{\frac{1 + 2t + t^2}{2t}}$$
Factor the top term: $$\sqrt{\frac{(1+t)^2}{2t}}$$
Take the square root: $$\frac {1+t}{\sqrt{2t}}$$
Divide top and bottom by $\sqrt t$:  $$\frac {\frac 1{\sqrt{t}} + \sqrt{t}}{\sqrt 2}$$
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