Final answer:
The greatest integer less than or equal to the cubic root of 504 is 7, as 7 cubed is 343 which is the largest perfect cube less than 504.
Step-by-step explanation:
The question asks for the greatest integer that is less than or equal to the cubic root of 504. To find this integer, start by looking for perfect cubes close to 504. The perfect cubes that are closest to 504 are <8>^3 = 512 and <7>^3 = 343. Since 343 is less than 504 and 512 is greater, the largest integer less than or equal to the cubic root of 504 is 7. Cubing 7 gives us 343, and since cubic roots deal with finding a number that when cubed gives the original number, the cubic root of 504 will be between 7 and 8. Since we're looking for the greatest integer less than or equal to this value, the answer is 7.