The prime factorization of 343 is 7^3. So the cube root of 343 (7^3) is just 7.
Irrational: Cannot be expressed in the form of a/b. 7/1 = 7 No
Rational: Can be expressed in the form of a/b. 7/1 = 7 Yes
Integer: Set of all negative and positive numbers (not a fraction). 7 is a positive number. Yes
Whole: Set of numbers ranging from 0 to infinity (not a fraction). 7 is a positive whole number. Yes
Natural: Set of all positive numbers from 1 to infinity (not a fraction). 7 is positive. Yes