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 $\frac{2}{2}=1$ .
 A: Your question has many analogies:
"If adding number to itself means multiplying it by 2, then why doesn't subtracting a number from itself mean dividing it by 2?" and so on
The problem you have noticed is not uncommon at all. It is the problem of specifying the logic that looks plausible for one particular case but that cannot be generalized. Try:
"If taking third power of a number means multiplying that number with itself two times then shouldn't taking a third root of a number mean (to do what) a number by itself?"
Although your question seems logical, you cannot extend it to any higher power.
You agree that by the squaring a square root, you need to come back to the original number. Obviously if we define it your way, you will not.
Multiplication and division are not the opposite operations. $a \cdot b = \frac{a}{b} \cdot b^2$ so $a \cdot a = \frac{a}{a} \cdot a^2$
There is indeed somewhere in the expression $ .. = \frac{a}{a}... $ but all other factors are missing.
When in doubt, try to generalize and you will notice why your logic is failing.
So:
"If taking n-th power of a number means multiplying that number with itself (n-1) times then shouldn't taking an n-th root of a number mean dividing a number by itself (n-1)-times?"
$a/a/a/a/a/a/... (n-1)-times .../a=a^{2-n}$ and $a^{2-n}$ is not n-th root of a number.
Well $a^{\frac{1}{x}}=a^{2-x}$ has a solution $x=1$ which means that your logic is valid for $x=1$ turning your question into:
"If taking first power of a number means do not multiply that number with itself at all, then shouldn't taking a first root of a number mean do not divide a number by itself at all."
For $n=1$ and for $n=1$ only it does mean that. But $x^1$ or $\sqrt[1]{x}$ are no valid examples of the power function or root function, since they do not contain their characterizations in any particular sense.
A: Squaring when explained in simple English, uses the word "itself". Here is an attempt to define the reverse process, finding square root, using the word "itself":
The square root of a number $N$ is  that number $x$ such that when $N$ is divided by $x$ it gives itself (my grammar is poor, subject and object of this sentence. But I hope you get the drift)
Edit: this idea translated to an equation would give the following:
if $N = 9$ then $x = 3$ and $N/x = 9$?? 
I guess itself in this context refers to $x$ and not $N$
A: Another explanation that I'm not seeing in the other answers:

The square root of a number $N$ ($\mathrm{4}$ in your example) is the number that if squared ($\mathrm{2}$) would come out to $N$ ($\mathrm{4}$).

Another way of thinking about an “inverse operation” (as others have noted) is a “what-if operation”.  For the square root in your example, you're asking “What if I had a number that when squared is $\mathrm{4}$?  What would my number be?”
Also, the name “square root” seems to be asking “what is the root of the square?”  In the form $R^2 = N$, the $R$ part is called the root of the operation and the $^2$ is the square operation itself.  Asking for the square root is asking “what root ($R$), if squared, would come out to our number ($N$)?”
A: You have to distinguish between ordinary language and technical language. For convenience, ordinary language is mixed with technical language, but you must guard against being misled by ordinary language. In particular, the ordinary-language term “opposite” is not well-defined. For example, is land travel the opposite of sea travel? or of air travel? So also in regard to your question: division is not the only “opposite” of multiplication. Consider the distributive property:
a(b + c) = ab + ac. Applied from left-to-right, the distributive property is called multiplication, but applied from right-to-left, it is called factoring. Thus FACTORING is also an “opposite” of multiplication, and, it so happens, the one that applies in this case - that is, finding the square root of a number means finding “two equal factors” for the number.
The worst (or, best) example of a disconnect between ordinary language and mathematical language concerns divisors of 0. According to the definition of a divisor, 23 is a divisor of 0, as is 37. Therefore, based on ordinary language, the real number system has divisors of 0, but according to mathematical language the real number system does NOT have divisors of 0.
Another good (bad) example of the disconnect between ordinary language and technical language is the difference in meaning between the formula for simple interest and the formula for compound interest: the formula for simple interest gives you exactly what it says, but the formula for compound interest gives you the total growth amount (so, to get the amount of compound interest, you have to subtract the principal from it). This example of the disconnect between ordinary language and technical language has the advantage of not requiring familiarity with ring-theoretic considerations.
Yet another good (bad) example of the disconnect between ordinary language and technical language is the fact that the naive reaction to hearing the phrase “the error involved in using the Trapezoidal Rule” is, “Well, if there is error involved in using the Trapezoidal Rule, then it’s pretty obvious that we shouldn’t be using it.”
Yet another good (bad) example of the disconnect between ordinary language and technical language is the defining of the notion of a “greatest” common divisor even though there is no (or, at least, not necessarily any) order relation defined (i.e., for an integral domain).
A: Being more specific while translating from prosaic languages to the math language helps.

I really like this question because for me it summarizes general problems of math teaching.
A: Since this question hinges directly on some fundamental ideas of math, this answer attempts to explicate those ideas in a similarly fundamental way.
Squaring a number can be thought of as a procedure.
The particular procedure for squaring a number can use a
template like the following:
$$ \Box \longrightarrow \Box\times\Box \longrightarrow \Box $$
We put the "input" value, for example, $2$, in the leftmost box, like this:
$$ 2 \longrightarrow \Box\times\Box \longrightarrow \Box $$
Next we make copies of the leftmost box and put them in the two
boxes in the middle:
$$ 2 \longrightarrow 2 \times 2 \longrightarrow \Box $$
Notice that these two boxes must each contain the same number.
Finally, we perform the indicated multiplication and write the result
in the last box on the right:
$$ 2 \longrightarrow 2 \times 2 \longrightarrow 4 $$
To take a square root, we want to reverse the procedure, that is,
work it backwards. So we take the "input" number, for example, $9$,
and put it in the box on the right:
$$ \Box \longrightarrow \Box\times\Box \longrightarrow 9 $$
Now we have to decide what to put in the two boxes in the middle.
We know we need the contents of the two boxes to be equal, and we know
that when we do the multiplication the result has to be $9$.
Suppose we guess the number in each box should be $3$. Then we have:
$$ \Box \longrightarrow 3\times3 \longrightarrow 9 $$
We can confirm that $3\times3$ does indeed give the result $9$, so all
is good so far. Now we just need to deduce what number was in the
leftmost box. We know the middle boxes were filled by copying that box,
so it had to contain a $3$ as well. So we have
$$ 3 \longrightarrow 3\times3 \longrightarrow 9 $$
And that's why the square root of $9$ is $3$ rather than $9/9$.
(Well, that and the fact that we refuse to put $-3$ in the two boxes in
the middle, because life is better when we consistently follow a rule
that says a "square root" must never be a negative number.)
We may later learn how to find square roots in a way that does not
rely so much on making a lucky guess. But that's a matter of an
algorithm for calculating a square root, not the definition of a square root.
A: Lets see, we have:


*

*square? ok then multiply

*square-root? ok then divide
So far so good. Now it seems the multiplication is by the number itself. We have:


*

*square? ok then use the number itself

*square-root? ok then use ...?
It seems that in general, operations (like multiplication) and inverse operations (like division) are only valid inverses of each other when dealing with a given reoccurring value. The exact same value will have to be a given operand to both operations. Therefore, in this context where there is no such reoccurring operand the inverse relationship is no more valid.
A: When you 'square' a number, you multiply a number by itself. But, when you derive the 'square root' of a number, you essentially find a number which, when 'squared', will give the number we're taking the 'square root' of. 
I think geometrical analogy can help you conceptualise. Think of 'squaring' as finding the area of a square with a certain length of sides while 'taking square root' will refer to finding the length of the sides of a square with a certain area. 
A: I'll take a slightly different approach here and say simply because:
That is not how square roots were defined.
Mathematics is entirely built on definitions. That's not a square root because it by definition is not a square root.
A: taking square root means reversing the effect of squaring. Dividing a number by itself does not do that (but rather always returns 1 as you noted).
Compare your question to: if doubling a number means adding it to itself, shouldn't halving a number mean subtracting it from itself? Answer: obviously not.
A: An explanation with units
. 
In short: taking the square  of  length $l$ yields the area $l^2$  of the square with side length  $l$. Taking the square root of something, interpreted as the area of a square $l^2$, yields the size of the side $l$.
Assume a side of length $3$ meters (3 m). Squaring transforms the side in a geometrical square of area $3\times3$ square meters (m$^2$). The name "square" is very consistent. And units are squared too! 
The  reverse operation consists in, starting from a square of a certain area given in (m$^2$), finding the  root, in other words the  side (in m) that would have produced "this" square. If you divide the area by itself, you should divide units too, and you end up with $1$, a unitless number, clearly not the answer (in m). 
You can better interpret this as different dimensions, combined as powers to numbers and units. $3$ meters are $3^1$ m$^1$ in one dimension. It yields a square of area $3^2$ m$^2$ in two dimensions. So, you somewhat multiply "powers" by $2$ when squaring. 
When taking the  root, you divide the power by two. A 9 square meters square is for instance $9^1$ m$^2$, so if you divide powers by two, you get the correct answer: $9^{\frac{1}{2}}$ m$^{\frac{2}{2}}$. 
Even if your problem is unitless, consider it has having $d$-unit" per dimension, and multiply and divide powers.
Another question is: can one imagine a square with negative sides? Because its area is the same as the one with positive sides. There are two square roots for $9$: $-3$ and $3$.
A: TL:DR? 'Root' has a specific meaning in mathematics. Taking the square root of a number means getting the 'root' of the equation that has the number on one side of the '=' sign, and a squaring operation on the other.

First, some terminology:


*

*In its most simple terms, a mathematical expression is a statement that can represent a single number.
eg. (5+6)x2, or (3+n)/(4+n) (where n represents some other number), etc.

*In the latter example, the expression contains a variable, n. We say that this expression is a 'function of n', which we can write as f(n) (or g(n), or h(n), or myfunction(n), etc.). In this example we may write: f(n) := (3+n)/(4+n) (Note that := means 'is defined to be'.)

*An equation is a statement of equality between two different mathematical expressions. eg. (5+6)x2 = 22, or (3+n)/(4+n) = 100, or 15-8=n.
If an equation contains one unknown, like the latter two examples above, then there should be one or more values that this unknown can take in order for the equation to be satisfied (i.e. for the equality to make sense). These are called the 'roots' of the eqaution.
By definition, the act of squaring a number, n, can be written as a function (let's call this 'square'). So, square(n) := n^2. 
If we know the answer to this, eg. 64, then we can write the equation:
 square(n) = 64,      or equivalently:      n^2 = 64.

The 'roots' of this equation are the values of n that fit. In this case, 8 and -8.
So: the SQUARE ROOT of a number X is the one and only positive ROOT, n, of the equation where X is equal to the square function applied to n: 
 square(n) = X.

(We take the positive root by convention.)
A: Consider these two functions:


*

*$s$ takes items of food as input, and puts a slice of lemon on top of each.

*$z$ takes drinks and adds 5% lemon juice to each.


Now suppose something has gone wrong and you need to reverse the procedure. Easy enough for $s$:


*

*$s^{-1}$ takes items of food as input, and takes away what's sitting on top.


This is a lot like inverting multiplication† by a constant factor: you have done some modification, and know that the result has some structure (like lemon sitting on top) so you can easily undo the multiplication, namely by division.
Not so in the second example. Here, the lemon juice has already mixed into the drink by the time you try to take it away. It's obviously no good to discard a spoonful from the surface of the drink. With the square root, you have an analogous problem: by self-multiplying a variable number, you forget the information of where exactly something was multiplied. You can't recognise a given shape of lemon slice that you could take off / divide away.

†Arguably, it's in fact more like addition and substraction than like multiplication and division. But those two pairs behave the same, for all that matters for this question (mathematically speaking: they both form groups).
A: When you square a number it flowers...
As you know when you square a number it gets really big, then bigger and bigger - you can think of it as branching out.

Conversely when you go "downwards" towards the root of a number, you are going down "inside" it. It gets dramatically smaller and smaller.
As you know multiplication and division are really just addition and subtraction. There is none of that branching power.

Setting aside mnemonics BTW the actual answer is simply:
In old fashioned language a 'root' means a 'solution'...
that's all there is to it.
In this case it's the "solution" to the "square".
So, x2 = 9, whats the "root" or "solution" to that equation.
(It's quite incredible only one answerer above pointed this out!)
A: This is the same as saying, if multiplying a number by 2 means adding it to itself, then shouldn't dividing a number by 2 mean subtracting it from itself? Same thing here.
A: Another way to express $\sqrt{x}$ is $x^\frac{1}{2}$.
(The 2 in the exponent comes from the fact that it's a square root. If it were a cube root, like $\sqrt[3]{x}$, it would be $x^{\frac{1}{3}}$.)
When you square a number, it's $x^2$; when you take the square root of a number, it's $x^\frac{1}{2}$. So we can see that the operations actually ARE opposites, just not in the way you think.       
A: 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 $z$.
A: The name square root comes from it being a root of this equation $x^2-A=0$ .
In this form it has nothing to do with division. In fact, we don't even need to know what division is, to formulate it.
Addition: 
So to explicitly answer the question - no, we should not mean to divide a number by itself when taking a square root, because it won't satisfy(solve) that equation. (except for A=1, to be entirely corect)
A: That is not a useful function because it always equals 1 (except 0/0).
Also, the opposite of $2\to4$ is $4\to2$, where you divide by the number you started with, not by 4.
A: Consider the following question, which is just yours with simpler operations substituted. Then you will hopefully understand.

If doubling a number means adding that number to itself, then shouldn't halving a number mean to subtract a number from itself?


And if you've got this, here is some more to contemplate. Suppose your boss proposes to raise your salary by any percentage you like, with as only condition that at the end of the year it will be lowered again by the same percentage; which percentage would you choose? The highlighted question is relevant to the choice "100%".
A: 
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 ?

In a certain sense, repeated multiplication $(a^n)$ and repeated division $(a^{-n})$ are indeed opposite operations to one another, pretty much for the same reason that simple multiplication and simple division are also opposites. But, from a different perspective, repeated multiplication $(a^n)$ and root extraction $\big(a^{1/n}\big)$ are also opposites. How so ? Notice that, in the first case, by multiplying the two quantities, we get $1$ as a result, which is the neutral element for multiplication; i.e., $a^n\cdot a^{-n}=1.$ In the second case, however, by composing the two operations, we get the argument a itself as a result, i.e., $(a^n)^{1/n}=\big(a^{1/n}\big)^n=a.$
A: What use is a term for dividing a number by itself?  The result is always $1$.
We define the square root as the (positive) number, that when multiplied by itself, gives the desired number.  This is a very useful expression. 
A: If squaring a number means multiplying it by itself, then taking the square root of a number means dividing the number by its square root.
Now as a definition that's a bit circular.  Unsquaring this circle happens to not be expressible with fundamental arithmetic operations.
A: As $\sqrt{n}\cdot\sqrt{n} = n$, then $\sqrt{n} = n\div\sqrt{n}$ and not $n\div n$
A: If you take a number, $N$ and multiply it times $x$ and then again times $x$, you would end up with $Nx^2$. To get $N$ back, you would divide by $x$ and then divide again by $x$.
The word root is generally used to represent one of the zeros of a function; that is a solution to $f(x) = 0$. In particular, a square root of N  is a solution to the polynomial equation $x^2 - N = 0$. If $N$ is a non negative real number, then there is a positive square root and a negative square root of $N$. Usually we express the positive square root of $N$ as $\sqrt N$ and the negative square root of $N$ as $-\sqrt N$.
ALSO
I suppose that , if you want dividing by $x$ in the definition of square root, then you can say that the square root of $N$ means finding the number, $x$, such that $\dfrac Nx = x$.
A comparison
\begin{array}{ccc}
   \text{action} & \text{equation} & \text{inverse} \\
\hline
   \text{doubling} & 2x = x + x & x = \dfrac 12(2x) \\
   \text{squaring} & x^2 = x \cdot x & x = \sqrt{x^2} \\
   \hline
\end{array}
Notice that the inverse of doubling is not subtracting x and that the inverse of squaring is not dividing by x.
