If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself? If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself? 
For example the square of $2$  is $2^2=2 \cdot 2=4 $ .
But square root of $2$ is not $\frac{2}{2}=1$ .

Assuming $x > 0$
Algebraically:
$$
\begin{matrix}
x \cdot x = x^2 & \rightarrow & \sqrt{x \cdot x} = \sqrt{x^2} & \rightarrow & \sqrt{x} \cdot \sqrt{x} = x \\
\downarrow &  &  &  & \downarrow \\
x^2 \div x = x & \rightarrow & \sqrt{x^2 \div x} = \sqrt{x} & \rightarrow & x \div \sqrt{x} = \sqrt{x}
\end{matrix}
$$
Visually (expanding on dkeck's answer):

Graphically:
Another way of wording the question is:
"If the curve for $y=x^2$ intersects with the line for $y = x \cdot z$ at $x=z$, then shouldn't the curve for $y=\sqrt{x}$ intersect with the line for $y=x \div z$ at $x=z$?"
See for yourself, where $z=2$:

$y = \sqrt{x}$ intersects with $y = x \div z$ at $x = z^2$, not at $x = z$. $y = \sqrt{x}$ intersects with $y = x \div \sqrt{z}$ at $x = z$.
The question comes from the very subtle logical error of confusing the function $\lambda x \space \space x \cdot x$ with the function $\lambda a \space \space a \cdot x$. The above graph calls attention to the difference between these functions.
If squaring a number meant multiplying that number by $z$, and square root were defined as the inverse of square, then yes, taking the square root of a number would mean dividing it by $z$.
