This rather elegant fact has been going round on Twitter:
— Cliff Pickover (@pickover) April 15, 2015
However, there are other examples of this phenomenon…
@pickover @Derektionary This has not been unknown to the dabbler in such matters. The number 3 and 3/8 behaves in a parallel fashion.
— Chris Maslanka (@ChrisMaslanka) April 15, 2015
In fact, pleasingly, we can construct an infinite number of facts like this.
Let’s say that we want to find positive integers A, B and C, such that:
√(A + B/C) = A √(B/C) 
We can also suppose that B < C (otherwise our mixed fraction would be a little odd) and that the highest common factor of B and C is 1, so that the fraction B/C is in its lowest form.
To find a relationship between A, B and C, we can square  and rearrange the equation to get the A terms on one side:
___________________________ ⇔ A + B/C = A² (B/C)
_______________________________⇔ A = (A² – 1) (B/C)
_______________________________⇔ A/(A² – 1) = B/C , A > 1 
We can use double implications (⇔) here since we know that everything is positive.
Now,  tells us that two fractions with positive integer numerator and denominator are equal. Both fractions are in their lowest form (B/C because we said so earlier, while A and (A²–––1) clearly have a highest common factor of 1). Therefore the numerators and denominators are equal:
 ⇔ A = B and A² – 1 = C (if A > 1)
This means that there is an infinite family of solutions of the form:
Any integer greater than 1 will do as a value for A, so we have:
and so on…