Infinite roots

Let $\Large \sqrt {m\sqrt{n \sqrt{m\sqrt {n \sqrt{\cdots}}}}}=x$ $\therefore \sqrt {m\sqrt{nx}}=x$ Squaring two times we get, $x^4=m^2n x \Rightarrow x^3=m^2n$ In the question we want to evaluate x when m=3, n=7 and vice versa. Hence required answer= $m^2n-n^2m=mn(m-n)=7\times 3\times(7-3)$ $~~~~~~~~~~~~\Large =\boxed {\color{#D61F06}{\boxed{84}}}$

Let $\displaystyle S=\sqrt{7\sqrt{3\sqrt{7\sqrt{\ldots\infty}}}}$ and

$\displaystyle P=\sqrt{3\sqrt{7\sqrt{3\sqrt{\ldots\infty}}}}$

Squaring both of these, we get:

$\displaystyle S^2=7P \ldots \{1\}$

and $\displaystyle P^2=3S \ldots \{2\}$

Multiplying both of these, we get:

$\displaystyle SP=\sqrt{21\sqrt{21\sqrt{21\sqrt{\ldots\infty}}}}$

$\displaystyle \Rightarrow SP^2=21\times SP$

$\displaystyle \Rightarrow SP=21 \ldots \{3\}$

Therefore, the required expression is:

$\displaystyle \Rightarrow S^3-P^3$

$\displaystyle \Rightarrow S\times \{S^2\} - P\times \{P^2\}$

From equations $\displaystyle \{1\}$ and $\displaystyle \{2\}\ldots$

$\displaystyle \Rightarrow S\times \{7P\} - P\times \{3S\}$

$\displaystyle \Rightarrow 4 \times SP$

Finally, from equation $\displaystyle \{3\} \ldots$

$\displaystyle \Rightarrow \boxed{84}$

if i may ask , what is the math program you use to write an equation?