What is the general rule to simplify a radical expression in a square root?

For example, sqrt(2sqrt(3)-4)

1 Answer
Jul 23, 2018

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 p, q, r > 0 and p^2-q^2r is a perfect square s^2 then:

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 p=-4, q=2 and r=3 we have:

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