Find the exact value (not a decimal approximation) of the continued square root SQRT(10+SQRT(10+SQRT10+...?

Answer 1

phi Φ

The nested radical at phi is the only answer I found that cannot be boiled down to a decimal. (See the link in the Source below, mathworld.com) phi is called the golden ratio.

You can find another answer at wikipedia, though the answer is:

( 1 + SQRT ( 1 + 4(10) ) ) / 2

(1 + SQRT(41))/2
[10 replaces any number that yo'ure using as n in SQRT(n + SQRT(n +..]

Of course, this can be mathed out somehing like: 3.70156211871....

http://en.wikipedia.org/wiki/Nested_radical#Infinitely_nested_radicals

Answer 2

Your "thing" satisfies to x^2= 10 + x , so = (1+sqrt{41})/2..