Cube root is a mathematical operation that is used to find the number which, when multiplied by itself three times, gives the given number. In this case, we are asked to find the cube root of 27000.

To solve this problem, we can use prime factorization.

Step 1: Prime Factorization of 27000
To find the prime factorization of 27000, we divide it by prime numbers until we can no longer divide evenly.

27000 ÷ 2 = 13500
13500 ÷ 2 = 6750
6750 ÷ 2 = 3375
3375 ÷ 3 = 1125
1125 ÷ 3 = 375
375 ÷ 5 = 75
75 ÷ 5 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1

So, the prime factorization of 27000 is 2^3 × 3^3 × 5^3.

Step 2: Taking Out the Cube Root
To find the cube root, we can take out one factor of each prime number from the prime factorization.

Taking out a factor of 2, we have: 2 × 2 × 2 = 2^3
Taking out a factor of 3, we have: 3 × 3 × 3 = 3^3
Taking out a factor of 5, we have: 5 × 5 × 5 = 5^3

So, the cube root of 27000 can be written as 2^3 × 3^3 × 5^3.

Step 3: Simplifying the Cube Root
We can simplify this expression by taking out one factor of each prime number.

2^3 × 3^3 × 5^3 = 2 × 3 × 5 × 2 × 3 × 5 × 2 × 3 × 5
= (2 × 3 × 5) × (2 × 3 × 5) × (2 × 3 × 5)
= 30 × 30 × 30

Therefore, the cube root of 27000 is 30.