The given three numbers are 5,8,24.
We have to use four operations (Adding, Subtracting,multiplication, division, square, square root, cube, cube root) out of these operation to make the result an irrational number.
There are many more examples that you can write.
Keep in mind
1. Sum of Rational and irrational is always irrational.
2. Difference of rational and irrational is always irrational.
As none of the 5,8,24 is a perfect square , nor addition of any two results in a perfect square, so square root of their addition or subtraction or multiplication results in an irrational number.
Similarly ,apart from 8, none of 5,24 is a perfect cube.So if you add or subtract any number from 8 and take their cube root or square root ,results in an irrational number.