Final answer:
The smallest non-zero whole number n such that 765n is a perfect cube is 9.
Step-by-step explanation:
To find the smallest non-zero whole number n such that 765n is a perfect cube, we need to factorize 756 and find the value of n.
Given that 756 equals 2 squared times 3 cubed times 7, we can rewrite it as 2^2 * 3^3 * 7.
For 765n to be a perfect cube, the powers of each prime factor in the factorization of 765n must be a multiple of 3.
Since the powers of 2 and 7 in 756 are already multiples of 3, we need to find the smallest value of n such that the power of 3 in 756 multiplied by n is a multiple of 3.
So, we have 3^3 * n = 3^k, where k is a multiple of 3.
By comparing the powers of 3 on both sides, we can see that k = 6.
Therefore, n = 3^(6/3) = 3^2 = 9.
So, the smallest non-zero whole number n such that 765n is a perfect cube is 9.