Simplifying Algebraic Expression Under A Square Root

Question

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.

Answer 1

Note that $$t + 2 + \frac{1}{t} = t + 2\cdot t\cdot \frac{1}{t}+ \frac{1}{t}= \left(\sqrt{t} + \frac{1}{\sqrt{t}}\right)^2$$

Answer 2

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 $\sqrt t$: $$\frac {\frac 1{\sqrt{t}} + \sqrt{t}}{\sqrt 2}$$