We can solve the by the method of contradiction.
Let us assume that 3✓2-5✓7 is a rational number.
A rational number is always in the form of p/q where p and q are intergers, p and q are co prime and q not equal to zero.
3✓2-5✓7 = p/q
Squaring both sides
193 - 30✓14= p^2/q^2
-30✓14= p^2/q^2 -193
Notice that LHS is irrational(14 is not a perfect square) but RHS is rational.
Hence this situation is not possible.