$\begin{align}  & \text{The square root of }19-4\sqrt{21} \\ & \text{Let }\sqrt{19-4\sqrt{21}}=\sqrt{a}-\sqrt{b}\text{ } \\ & \text{(we need the positive square root)} \\ & \text{Square both sides of the equation} \\ & 19-4\sqrt{21}=a+b-2\sqrt{ab} \\ & \text{Comparing terms} \\ & a+b=19 \\ & b=19-a----(i) \\ & 2\sqrt{ab}=4\sqrt{21} \\ & ab=84----(ii) \\ & \text{substitute }b\text{ into equation }(ii) \\ & (19-a)a=84 \\ & {{a}^{2}}-19a+84=0 \\ & (a-12)(a-7)=0 \\ & a=12\text{ or }a=7 \\ & \text{Since we need the positive square root} \\ & b=19-12=7 \\ & 19-4\sqrt{21}=\sqrt{12}-\sqrt{7}=2\sqrt{3}-\sqrt{7} \\\end{align}$