This is a PDF, not a PMF, and so we know that "days" is a continuous number. To transform PDF into CDF, integrate from -infinity (in this case 0 is fine, since it's the lowest value for which the PDF is defined) to the upper number of days; in this case, 364 (since the problem is asking "less" than 365).
Integral of a constant times a function = constant times the integral of a function, so take (5/(500^5)) out of the function.
Int(0 to 364) x^4 should be trivial to you, then take the constant back in to cancel out the 5s, and you get the following for the definite integral:
364^5 / 500^5 - 0 = ~0.2045
I think the hint gives you what you need.
f(x) = 5x^4/(500^5) <-- original PDF for single component failure
F(X <= x) = x^5/(500^5) <--- integration of f(x) to go from PDF to CDF
F(Y <= y) = (y^5/(500^5))^3 = (y^15)/(500^15) <-- transform from single component failure to three-in-parallel failure considering their hint
f
= d/dy F
= 1/(500^15) * d/dy y^15 = 15y^14 / (500^15) <-- recovering three-in-parallel PDF given three-in-parallel CDF
I believe your error is that you were trying to cube the PDF when you should have been cubing the CDF and then recovering the resulting PDF, am I correct?