Yes, there is a number that satisfies the equation where a number is exactly 5 more than its cube. Let’s denote the number as x. We can set up the equation:
x^3 = x + 5
This implies:
x^3 - x - 5 = 0
To find solutions to this equation, we can try to see if there are any rational roots using the Rational Root Theorem. According to the theorem, if there is a rational root, it must be a factor of the constant term (in this case, -5). Thus, the possible rational roots could be: ±1, ±5.
Let’s evaluate these possible roots:
- For
x = 1:1^3 - 1 - 5 = 1 - 1 - 5 = -5(not a root) - For
x = -1:(-1)^3 - (-1) - 5 = -1 + 1 - 5 = -5(not a root) - For
x = 5:5^3 - 5 - 5 = 125 - 5 - 5 = 115(not a root) - For
x = -5:(-5)^3 - (-5) - 5 = -125 + 5 - 5 = -125(not a root)
Since none of the rational roots work, we can also use numerical methods or graphing techniques to find an approximate root. Using these methods, we can find that there is a real number solution around x ≈ 1.5.
To check this, we plug x = 1.5 back into the original condition:
(1.5)^3 ≈ 3.375 and 1.5 + 5 = 6.5. Since 3.375 is not equal to 6.5, x = 1.5 is not a solution, but it shows that you can find numbers near this value that approach the equality.
In conclusion, while there is no integer solution that satisfies the equation, there are real numbers that come close, and this highlights the interesting relationship between numbers and their powers.