By definition, the cubic root of a number x is a number y such that y^3=x.
Apart from using the calculator, of course, you can see if a number n is a perfect square by factoring it into primes, and if the number has a representation of the form
n= p_1^{d_1} \times p_2^{d_2}\times ... \times p_n^{d_n}, then it is a perfect cube if and only if every d_i is divisible by 3.
Factoring 128 in primes gives you
128=2^7, thus it is not a perfect cube (i.e., its cube root is not an integer).
Anyway, we can say that the cubic root of 128 is 128 to the power of 1/3, so we have
128^{1/3}=(2^7)^{1/3}=2^{7/3}=2^{2+1/3}
By using the formula a^{b+c}=a^b\cdot a^c, we have that
2^{2+1/3}=2^2 \cdot 2^{1/3}=4 \cdot 2^{1/3}
which is four times the cubic root of 2
