Final answer:
The number 160,000 has 8 perfect cube divisors, which are determined by counting the possible combinations of its prime factors' exponents that are multiples of three.
Step-by-step explanation:
To determine how many perfect cube divisors the number 160,000 has, we first need to factor this number into its prime factors. The prime factorization of 160,000 is:
160,000 = 29 × 55.
For a number to be a perfect cube, its prime factors must be to the power of a multiple of 3. Therefore, to find the perfect cube divisors, we count the number of ways we can create exponents of 2 and 5 that are multiples of 3.
For 29, the exponents that are multiples of 3 are 0, 3, 6, and 9. For 55, the exponents that are multiples of 3 are only 0 and 3, since 56 would exceed the prime factorization of 160,000.
Thus, there are 4 × 2 = 8 different combinations for the exponents, meaning there are 8 perfect cube divisors of 160,000.