I thought it was correct because sqrt is the opposite of multiplying by a number, so I figured by multiplying by a number it would balance out and be that number normally, but when I tried it with my Python calculator using 3 I got:

invernessgangshow.net.sqrt(3) * 3 5.196152422706632

$\begingroup$ $\sqrt x$ is the number satisfying $\sqrt x \sqrt x = x$; it's not the opposite of multiplying by $x$. $\endgroup$
In general, no. Since $\sqrt x$ is equal to $x^{1/2}$, your equation is the same as $x^{3/2} = x$, only $x = 0, 1$ work as solutions

If you want to solve for the equation$$x\sqrt{x} = x$$then you have$$x (\sqrt{x}-1) =0 \Rightarrow x = 0, \text{ or } \sqrt{x}=1, \text{ i.e. } x = 1$$

The multiplicative inverse (the "opposite" in your question) of a non-zero number $x$ is its reciprocal, $\frac{1}{x}$. Multiplying these two together gives instead $x \times \frac{1}{x} = 1$.

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calculate the the limit of the sequence $a_n = \lim_{n \to \infty} n^\frac{2}{3}\cdot ( \sqrt{n-1} + \sqrt{n+1} -2\sqrt{n} )$