<s>[INST] Simplify $\left(\sqrt{\left(\sqrt{2} - \frac{3}{2}\right)^2} - \sqrt[3]{\left(1 - \sqrt{2}\right)^3}\right)^2$ I was trying to solve this square root problem, but I seem not to understand some basics. 
Here is the problem.
$$\Bigg(\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\bigg)^2} - \sqrt[3]{\bigg(1 - \sqrt{2}\bigg)^3}\Bigg)^2$$
The solution is as follows:
$$\Bigg(\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\bigg)^2} - \sqrt[3]{\bigg(1 - \sqrt{2}\bigg)^3}\Bigg)^2 = \Bigg(\frac{3}{2} - \sqrt{2} - 1 + \sqrt{2}\Bigg)^2 = \bigg(\frac{1}{2}\bigg)^2 = \frac{1}{4}$$
Now, what I don't understand is how the left part of the problem becomes:
$$\frac{3}{2} - \sqrt{2}$$
Because I thought that $$\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\bigg)^2}$$
equals to $$\bigg(\bigg(\sqrt{2} - \frac{3}{2}\bigg)^2\bigg)^{\frac{1}{2}}$$
Which becomes $$\sqrt{2} - \frac{3}{2}$$
But as you can see I'm wrong. 
I think that there is a step involving absolute value that I oversee/don't understand. 
So could you please explain by which property or rule of square root is this problem solved? 
Thanks in advance
 [/INST] Nicely put question.
You are right about the absolute value missing somewhere. Indeed, we have:
$$\sqrt{x^2} = |x|.$$
In your case, we have
$$\sqrt{\left(\sqrt{2}-\frac{3}{2}\right)^2}=\left|\sqrt{2}-\frac{3}{2}\right|.$$
But $\sqrt{2}-\frac{3}{2}$ is negative, so the absolute value "chooses" the positive version of this, that is,
$$\left|\sqrt{2}-\frac{3}{2}\right| = -\left(\sqrt{2}-\frac{3}{2}\right)=\frac{3}{2}-\sqrt{2}.$$
I hope this helps.
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