Final answer:
To express (6 1/4)^3 in simplest radical form, convert the mixed number to 25/4, then cube the numerator and denominator separately resulting in 15625/64, which simplifies back to 25/4 because both are perfect cubes.
Step-by-step explanation:
To express (6 1/4)^3 in simplest radical form, we convert the mixed number to an improper fraction and then apply the cube operation. First, convert 6 1/4 to an improper fraction: 6 1/4 = (6×4 + 1)/4 = 25/4. Thus, (6 1/4)^3 = (25/4)^3.
When you cube an improper fraction, you cube both the numerator and the denominator separately. So, (25/4)^3 = 25^3/4^3 = 15625/64. This fractional form can't be simplified further, but if you wish to express it in radical form, it will be the cube root of 15625 over the cube root of 64, which is cube root of 15625/cube root of 64. However, since 15625 and 64 are both perfect cubes (25^3 and 4^3, respectively), the answer simplifies to 25/4.