What is the smallest number by which 2,160 should be divided to make it a perfect cube ?

Sure, here are the steps to find the smallest number by which 2160 should be divided to make it a perfect cube:

1. First, we need to find the prime factors of 2160. The prime factors of 2160 are 2, 2, 2, 2, 3, 3, 3, 5.

2. To form a perfect cube from the factors, each factor must appear three times (because a cube is a number multiplied by itself twice, i.e., to the power of 3).

3. Looking at the factors, we see that 2 appears four times and 3 appears three times, but 5 appears only once.

4. Therefore, to make 2160 a perfect cube, we need to remove one instance of 2 (to make it appear three times) and two instances of 5 (to make it appear three times).

5. So, the smallest number by which 2160 should be divided to make it a perfect cube is 255 = 50.