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From compound interest to bubonic plague, things that grow or spread really fast are often modeled by exponential functions. Learn about these powerful functions (pun intended?).

What is the sum of all integer values of \(x\) such that

\[ (x^2-17x+71)^{(x^2-34x+240)} =1 ? \]

**Details and assumptions**

\(x\) can be a negative integer. Since the exponent is an integer, the value is still well defined.

\(0^0\) is undefined.

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\[\Large a^{(a-1)^{(a-2)}} = a^{a^2-3a+2}\]

Find the sum of all positive integers \(a\) that satisfy the equation above.

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\[ \Large \text{ If } x^{x^{x^{16}}} = 16, \text{ then evaluate } x^{x^{x^{12}}}. \]

Note: \(x\) is a real number.

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In England, with respect to the initial population each year, the death rate is \(\frac{1}{46}\) and the birth rate is \(\frac{1}{33}.\)

If there were no emigration, how many years would it take for the population to double?

Note: Round up your answer.

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We know
\[a^{m+n}=a^m \times a^n\]
But there are some students who claim that
\[a^{m+n}=a^m+a^n\]
which is obviously incorrect. However, it is true for some triplets of integers, \((a,m,n)\). If \(a\in[0,10]\) and \(m,n\in[1,10]\), then find the number of possible triplets for which \(a^{m+n}=a^m+a^n\).

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