If ${a^x} = {b^y} = {(ab)^{xy}}$,then $x + y = $

The value of ${81^{(1/{\log_5}3)}} + {27^{\log_9 36}} + {3^{4/{\log_7}9}}$ is

Let $p = 99$ and $q = 101$. Define $p_1 = \log_{10} \left(\frac{p+q}{2}\right)$ and $q_1 = \frac{1}{2}(\log_{10} p + \log_{10} q)$,and $p_2 = \log_{10} \left(\frac{p_1+q_1}{2}\right)$,$q_2 = \frac{1}{2}(\log_{10} p_1 + \log_{10} q_1)$. Then:

The rational form of the number $1.\overline{41}$ is

If ${2^x} = {4^y} = {8^z}$ and $xyz = 288$,then find the value of $\frac{1}{{2x}} + \frac{1}{{4y}} + \frac{1}{{8z}}$.

The value of $a^{\log_b x}$,where $a = 0.2$,$b = \sqrt{5}$,and $x = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots$ to $\infty$ is: