Since Pi is an infinitesimal number there is no exact square root, just as there is no exact number for Pi.
However we can get close enough for government work, or any other for that matter.
Since Pi is an infinitesimal number there is no exact square root, just as there is no exact number for Pi.No, it is not infintesimal. It is irrational.
Proof:
Suppose, by way of contradiction, that 
is rational. Then 
must also be rational. Thus 
where a,b 
Z, b 
0. Since 
, it follows that there exists an M > 0 such that if n 
M, 
, or 
.
Choose an integer N 
M. It follows then that 
. Define a polynomial 
. Expanding the polynomial one finds that 
, where each 
is an integer. For integers k such that 
, one also finds that 
. Now notice that 
since 
and 
. Now suppose this is true for an integer k, i.e. 
. Now
.
So the formula is true for all nonnegative integers by the principle of mathematical induction. Now for values of k such that 
each term in the expansion of 
and 
. Now for values of k such that 
the only term which does not contain a positive integer power of x is the first. This is the case when n=k. This means that 
, an integer since 
. It follows then that 
, also an integer.
Now define another function
.
It now follows that 
and 
are both integers and thus that 
is an integer. Now define 
. It follows then that
.
.
Now
, and so
. Also
.
Notice that since f(x) is a polynomial of degree 2N, 
for all x. Thus
. So
.
Now since g(x) is continuous on 
, and 
exists on 
, by the Mean Value Theorem, there exists a c 
such that 
. Now 
, and
. So 
, and thus
.
.
Since 
and also 0 < 1-c < 1. Since 
, it follows that 
.
It now follows that 
and so 
. Since 
is an integer, there must exist an integer between 0 and 1.
Thereby, reaching a contradiction, the theorem is proved. Q.E.D.
"However we can get close enough for government work, or any other for that matter."
Heck, close enought for gov't work would be ...oh...like... 12?
Its an irrational number, so it belongs to the democratic party, not an infinate decimal number.
" However we can get close enough for government work, or any other for that matter." Conceive a sphere constructed with the earth at it center, and imagine it surface to pass through Sirius, which is 8.8 light years distant from the earth... Then imagine this enormous sphere to be so packed with microbes that in every cubic millimeter millions and millions of these diminutive animalcula are present. Now conceive these microbes to be unpacked and so distributed singly along a straight line that every two microbes are as far distant from each other as Sirius is from us... Conceive the long line thus fixed by all the microbes at the diameter of a circle, and imagine its circumference to be calculated by multiplying it diameter by Pi to 100 decimal places. Then, in the case of a circle of this enormous magnatude even, the circumference so calculated would not vary from the real circumference by a millionth part of a millimeter. This example will suffice to show that the calculation of Pi to 100 or 500 decimal places is wholly useless. - Hermann Schubart, A mathematics professor from Hamburg, Germany in 1889