<s>[INST] How can I solve $x^\sqrt{y} +y^\sqrt{x} =\dfrac{49}{48}$ and $\sqrt{x}+\sqrt{y} =\dfrac72$? I have tried Wolfram Alpha and Mathematica to get the solution of the below system, but no result , I have used variable change $z=\sqrt{x}+\sqrt{y}$ for simplification but no result , 
$$
\left\{
\begin{array}{ll}
x^\sqrt{y} +y^\sqrt{x} &=\dfrac{49}{48} \\ 
\sqrt{x}+\sqrt{y} &=\dfrac72 \\
\end{array} 
\right.
$$
How I can solve this?
 [/INST] Doing the same as Jack D'Aurizio (eliminating $y$ from the second equation) and using a Taylor expansion around $x=0$, the first equation is
$$-\frac{1}{48}+\sqrt{x} \log \left(\frac{49}{4}\right)+x \left(\frac{1}{2} \log
   ^2\left(\frac{49}{4}\right)-\frac{4}{7}\right)+O\left(x^{3/2}\right)$$ the solution of which being
$$x=\frac{7}{96 \left(-2+343 \log ^2\left(\frac{7}{2}\right)+4 \log \left(\frac{7}{2}\right) \sqrt{7350 \log
   ^2\left(\frac{7}{2}\right)-84} \right)}\approx 0.0000679848$$ Using this as a starting guess, Newton iterates would be 
$$\left(
\begin{array}{cc}
 n & x_n \\
 0 & 0.000067984814185381173107 \\
 1 & 0.000067980800578179758426 \\
 2 & 0.000067980800636422696813 \\
 3 & 0.000067980800636422696826
\end{array}
\right)$$ which is the solution for twenty significant figures.
</s>