Question

x is a number between 3 and 50.
How many vales of x is radical x / 13 a rational number?

Answer 1

"x is a number between 3 and 50." requires any number x (integer, fractional or decimal) between the given limits such that √(x/13) is rational.

The number of integers is limited to 1, namely 13, since √(13/13)=1 is rational. Multiplying 13 by the nearest perfect square exceeds the upper limit.

However, there are many non-integer rational numbers that fit the bill.

Basically, we multiply 13 by fractions whose numerator and denominator are perfect squares.

13/4
13*4/9
13*9/4
13*25/9
13*9/25
13*25/36
13*36/25
13*36/49
13*49/36
13*49/64
13*64/49
...
As you can see, the list can go on and on, and the list is long, if not infinite.

You may want to double check if the question actually meant:
"x is an integer between 3 and 50."

Answer 2

for √x/13 to be rational, x must be a perfect square.

perfect squares between 3 and 50 are

4,9,16,25,36, and 49