 Algebra

# Exponential Functions: Level 4 Challenges

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.

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

Find the sum of all positive integers $a$ that satisfy the equation above. $\Large \text{ If } x^{x^{x^{16}}} = 16, \text{ then evaluate } x^{x^{x^{12}}}.$

Note: $x$ is a real number.

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? 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 $a^{m+n}=a^m+a^n$.