What is the smallest number by which 288 must be multiplied so the product is a perfect cube?

Sure, here are the steps to find the smallest number by which 288 must be multiplied so the product is a perfect cube:

Step 1: Prime factorize the number 288. The prime factors of 288 are 2, 2, 2, 2, 2, 3, 3.

Step 2: Group the prime factors into triples. A perfect cube has all prime factors in groups of three. In this case, we can make two groups of 2 (2, 2, 2) and one group of 3 (3, 3), but we have one 2 and one 3 left over.

Step 3: The smallest number by which 288 must be multiplied to make it a perfect cube is the product of the leftover prime factors. In this case, that's 2 * 3 = 6.

So, the smallest number by which 288 must be multiplied so the product is a perfect cube is 6.