Think outside the box (3)

Calculus Level 4

Let \(f^{(0)}(x) = (5x^3+99x^{98}+2)^{100}\), where \(f^{(n)}(x)\) denotes the \(n^\text{th}\) derivative of \(f(x)\). If the value of \(f^{(98)}(0)\) is in the form \((a^2)!\times b^{99} \), where \(a\) and \(b\) are natural numbers . Find the value of \(a+b\).

\[\] Notation: \(!\) is the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).


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