As a math student, it is common to come across perplexing problems that require thorough thinking and analysis to solve. One such problem is determining the cube root of 64. At first glance, one may think that it is a simple problem with an answer of 4. However, that is only the square root of 64, not the cube root.

The cube root of a number is the value that when multiplied by itself thrice, gives the original number. In other words, the cube root of 64 is a value that when multiplied by itself three times, equals 64.

So, how do we determine the cube root of 64? The simplest method is through prime factorization. We express 64 as a product of prime factors, which gives us 2 raised to the power of 6 (2x2x2x2x2x2 = 64).

Since we are looking for a value that when multiplied by itself thrice equals 64, we can group the prime factors in sets of 3. We end up with 2×2 as one set and another 2×2 as the second set. When we multiply these two sets, we get 2x2x2 which equals 8. Therefore, the cube root of 64 is 8.

In conclusion, solving the mystery of the cube root of 64 requires critical thinking and understanding of prime factorization. By grouping the prime factors in sets of 3, we can easily find the cube root of any number.