Is the square root of -11.2 a rational number?

No, it is an imaginary number.
Can't take the square root of a negative number without creating a method termed an imaginary number, which is the square root of -1 which they have used " i " to signify.
Then square root of -11.2 would be written Sqrt(11.2)i.  The Sqrt(11.2) = 3.3466
so the answer is 3.3466i