Simplify the nested radical sqrt(11 - sqrt(120)) into a difference of two square roots.

Step-by-step solution

1. 1

   Put the inner radical in the form $2\sqrt{k}$. Denesting formulas expect a factor of $2$ in front of the inner root, so extract the perfect square from $120$:

   $\sqrt{120}=\sqrt{4\cdot 30}=2\sqrt{30}\;\Longrightarrow\;11-\sqrt{120}=11-2\sqrt{30}$

2. 2

   Set up the perfect-square template. If $11-2\sqrt{30}=\left(\sqrt m-\sqrt n\right)^2$ then expanding the right side gives

   $\left(\sqrt m-\sqrt n\right)^2=m+n-2\sqrt{mn}$

   so matching the rational and irrational parts separately requires

   $m+n=11,\qquad mn=30$

3. 3

   Solve the small system. These are the sum and product of two numbers, so $m$ and $n$ are the roots of $t^2-11t+30=0$:

   $t=\frac{11\pm\sqrt{121-120}}{2}=\frac{11\pm 1}{2}\;\Longrightarrow\;t=6\ \text{or}\ 5$

   The discriminant coming out as a perfect square ($1$) is precisely the condition for the radical to denest at all — most nested radicals do not.

4. 4

   Take the square root with the correct sign. Since a square root is non-negative, choose the ordering that makes the difference positive, i.e. $m=6$ and $n=5$:

   $\sqrt{11-\sqrt{120}}=\sqrt{\left(\sqrt6-\sqrt5\right)^2}=\left|\sqrt6-\sqrt5\right|=\sqrt6-\sqrt5$

5. 5

   Check numerically, and note the rationalized reciprocal. Numerically $\sqrt{120}=10.9545$, so $11-\sqrt{120}=0.0455488$ and its square root is $0.213422$; meanwhile $\sqrt6-\sqrt5=2.449490-2.236068=0.213422$ $\checkmark$ — the two agree exactly, with no gap to explain away. A useful companion fact: the reciprocal rationalizes neatly, since

   $\frac{1}{\sqrt6-\sqrt5}=\frac{\sqrt6+\sqrt5}{6-5}=\sqrt6+\sqrt5$

   so the radical and its reciprocal add to $2\sqrt6$.