What is the general rule to simplify a radical expression in a square root?
For example, sqrt(2sqrt(3)-4)
For example,
1 Answer
In your example, we find:
sqrt(2sqrt(3)-4) = (sqrt(3)-1)i
Explanation:
I explored this in https://socratic.org/s/aSS7FqaZ finding:
If
sqrt(p+qsqrt(r)) = sqrt(2p+2s)/2+sqrt(2p-2s)/2
If these conditions break down then we might expect something like:
sqrt(p+qsqrt(r)) = sqrt(2p+2s)/2-sqrt(2p-2s)/2
or:
sqrt(p+qsqrt(r)) = -sqrt(2p+2s)/2+sqrt(2p-2s)/2
In particular, with
s = sqrt(p^2-q^2r) = sqrt(16-12) = 2
sqrt(2p+2s)/2 = sqrt(-8+4)/2 = sqrt(-4)/2 = i
sqrt(2p-2s)/2 = sqrt(-8-4)/2 = sqrt(-12)/2 = sqrt(3)i
In your example the radicand is negative, but we can simplify by splitting the radicand into a perfect square:
sqrt(2sqrt(3)-4) = sqrt(-(3 - 2sqrt(3)+1))
color(white)(sqrt(2sqrt(3)-4)) = sqrt(-(sqrt(3)-1)^2)
color(white)(sqrt(2sqrt(3)-4)) = (sqrt(3)-1)i