Answer: False
Here are some counter examples to disprove the original claim.
- log(5) = 0.69897 ... where the log base is not any power of 5
- sin(27) = 0.45399
- pi = 3.14159
- e^3 = 20.0855369
The decimal values are approximate.
All of those values are known as transcendental numbers. These are a class of numbers where we cannot write them as a root of some other number. It's similar to the idea how irrational numbers cannot be written as a ratio of two integers. All transcendental numbers are irrational, but not the other way around.
So in short, it is possible to have an irrational number where it's not a radical.