Problem 3 · Use a common power for squares and cubes · 3 points
Made by AI and not reviewed by a real person.
A positive whole number has both a whole-number square root and a whole-number cube root. If those roots add up to 36, what is the number?
- 64
- 729
- 4096
- 1296
- 46656
Show a hint
- Think about how the prime factors must be grouped for both roots to be whole numbers.
- The number must be a sixth power, k⁶. What do its two roots add up to in terms of k?
Show the answer and solution
Answer: B — 729
Write the number as k⁶. Its square root is k³ and its cube root is k², so k³ + k² = 36. For k = 1, 2, and 3, the sums are 2, 12, and 36; larger k give larger sums. Thus k = 3 and the number is 3⁶ = 729. The tempting 1296 is 36², but its cube root is not a whole number.