The Iterative Degree

Instead of letting $P$ = $a_{1}x^{2} + b_{1}x + c_{1}$ and $f(x)$ = $a_{2}x^{4} + b_{2}x^{3} + c_{2}x^{2} + dx + e$ , we could let $P$ = $x^{2}$ and $f(x)$ = $x^{4}$ , since we only want to find the degree after they are manipulated.

Now,

$f(x)$ = $x^{4}$

$f^{(2)}(x)$ = $(x^4)^4$ = $x^{4^2}$

$f^{(3)}(x)$ = $(x^{4^2})^4$ = $x^{4^3}$

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$f^{(n)}(x)$ = $x^{4^n}$ , and this is what we will use to solve the problem.

Here is the one we have to solve:

$f^{(19)}(P \cdot f^{(19)}(P$ ))

= $f^{(19)}(x^2 \cdot f^{(19)}(x^2$ ))

= $f^{(19)}(x^2 \cdot (x^2)^{(4^{19})})$

= $f^{(19)}(x^2 \cdot (x^2)^{(2^{38})})$

= $f^{(19)}(x^2 \cdot (x^{2^{39}}))$

= $f^{(19)}(x^{2 + 2^{39}})$

= $(x^{2 + 2^{39}})^{4^{19}}$

= $(x^{2 + 2^{39}})^{2^{38}}$

= $x^{2^{39} + 2^{77}}$

Thus, $a = 2$ , $b = 39$ , $c = 77$ , and $a + b + c$ $=$ $2 + 39 + 77$ $=$ $\boxed{118}$