If ​, find x.

if you square it,

 x+ \(\sqrt{x+\sqrt{x+...}}\)=81

 

so x=72

Since the square root goes on for infinity:

\(\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x...}}}}=\sqrt{x+\sqrt{x+\sqrt{x...}}}\)

 

Let us set \(\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x...}}}}=y\)

 

This means: \(y=9\)

 

This also means: \(\sqrt{x+y}=9\)

Substitute \(y\): \(\sqrt{x+9}=9\)

Square both sides: \(x+9=81\)

 

\(\boxed{x=72}\)