Find the positive integer such that when 3 is added to its cube root and the result is cubed again, the new value is 513 greater than the original number.
216
Let the number be \(n^3\), so its cube root is \(n\).
Adding 3 to the cube root gives \(n+3\), and cubing this: \((n+3)^3 = n^3 + 513\).
Expanding, \(n^3 + 9n^2 + 27n + 27 = n^3 + 513\), which gives \(9n^2 + 27n - 486 = 0\), or \(n^2 + 3n - 54 = 0\).
Factoring, \((n-6)(n+9) = 0\), so the positive value is \(n = 6\).
So the original number is \(n^3 = 6^3 = 216\).
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