<s>[INST] Why can I square both sides? I am not used to English. I ask for your understanding in advance.
There is the equation:
$ x= 2^\frac{1}{2}$
we can square both side like this:
$ x^2= 2$
But I don't understand why that it's okay to square both sides.
What I learned is that adding, subtracting, multiplying, or dividing both sides by the same thing is okay. For example:
$ x = 1 $
$ x-1 = 1-1 $
$ x-1 = 0 $
$ x \times 2 = 1 \times 2 $
$ 2x = 2 $
like this.
But how come squaring both sides is okay too?
$ x = 2 $
$ x \times 2 = 2 \times 2 $
$ 2x = 4 $
$ 2x \times x = 4 \times x $
This does not induce it.
Can you answer this silly question?
 [/INST] We start from the fact that if $x = y$, then $a\cdot x = a\cdot y$ and $x\cdot a = y\cdot a$ for any $a$.
Therefore, if $x = y$, letting $a = x$ we get $x\cdot x = x\cdot y$ and letting $a = y$ we get $x\cdot y = y\cdot y$, so $$x^2 = x\cdot x = x\cdot y = y\cdot y = y^2.$$
Thus, we have shown that given equality $x = y$ it is ok to square both sides, i.e. $x = y$ implies $x^2 = y^2$.
</s>