Question 1106487
<br>
x = sqrt(7/3+sqrt(7/9+sqrt(7/3+sqrt(7/9+...))))<br>
x^2 = 7/3 + sqrt(7/9+sqrt(7/3+sqrt(7/9)...)))<br>
x^2-7/3 = sqrt(7/9+sqrt(7/3+sqrt(7/9)...)))<br>
(x^2-7/3)^2 = 7/9+sqrt(7/3+sqrt(7/9)...))<br>
(x^2-7/3)^2 = 7/9 + x<br>
x^4-(14/3)x^2+49/9 = 7/9 + x<br>
x^4-(14/3)x^2-x+42/9 = 0<br>
x^4-(14/3)x^2-x+14/3 = 0<br>
3x^4-14x^2-3x+14 = 0<br>
This polynomial has two real roots, x=1 and x=2, and a pair of complex roots.<br>
Clearly the given expression is real, because it is the square root of a positive number.  And clearly the expression is not equal to 1.  Therefore the value of the expression is 2.