| Another Radical Expression |
This problem was submitted by Sasha Joseph, this year's high scoring sophomore at the MAML state math meet.
If f(x) = 3b2x, with b a constant greater than zero, find the value of the following expression in terms of b.
 |
Problem Moderated by: Douglas |
| Problem Solution |
In order to solve this problem, first we recognize that the b's can be factored out of the entire expression. This is because the exponent of b is continually doubling within the nested radicals.
The result is:
b*sqrt(3 - sqrt(3 + sqrt(3 - ...)...)
Some solvers stopped here; however, this is not completely simplified.
Let X = sqrt(3 - sqrt(3 + sqrt(3 - ...)...)
X = sqrt(3 - sqrt(3 + X))
X2 = 3 - sqrt(3 + X)
X2 - 3 = sqrt(3 + X)
X4 - 6X2 + 9 = 3 + X
X4 - 6X2 - X + 6 = 0
Factoring this gives:
(X-1)(X+2)(X2-x-3) = 0
By testing the various roots, we can see that X=1 is the correct value.
Thus, the answer is 1b = b |
|
|
