If $12^{12x} \times 4^{24x + 12} \times 5^{2y} = 8^{4z} \times 20^{12x} \times 243^{3x - 6}$ where $x, y,$ and $z$ are natural numbers, then find the value of $x + y + z$.

Find the sum of all real values of $k$ for which $\left(\frac{1}{8}\right)^k \times \left(\frac{1}{32768}\right)^{\frac{1}{3}} = \frac{1}{8} \times \left(\frac{1}{32768}\right)^{\frac{1}{k}}$.

Let $a, b, m$ and $n$ be natural numbers such that $a > 1$ and $b > 1$. If $a^m b^n = 144^{145}$, then find the largest possible value of $(n - m)$.

If $x = 4096^{7+4\sqrt{3}}$, then which of the following expressions evaluates exactly to $64$?

If $a, b, c$ are non-zero real numbers and $14^a = 36^b = 84^c$, then find the value of $\frac{6b}{c} - \frac{6b}{a}$.

If $5.55^x = 0.555^y = 1000$, then find the value of $\frac{1}{x} - \frac{1}{y}$.

Given that $x^{2018}y^{2017} = \frac{1}{2}$ and $x^{2016}y^{2019} = 8$, find the value of $x^2 + y^3$.

Given that $x^{2018}y^{2017} = \frac{1}{2}$ and $x^{2016}y^{2019} = 8$, find the value of $x^2 + y^3$.

Find the value of the expression $\sqrt{1 + \frac{1}{1^2} + \frac{1}{2^2}} + \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} + \dots + \sqrt{1 + \frac{1}{2007^2} + \frac{1}{2008^2}}$.

If $x = \sqrt{7 + \sqrt{7 + \sqrt{7 + \dots}}}$, then which of the following expressions defines the boundary layout of $x$?