$\displaystyle \begin{array}{{>{\displaystyle}l}} \sqrt{\frac{A+\sqrt{A^{2} -B}}{2}} \pm \sqrt{\frac{A-\sqrt{A^{2} -B}}{2}} \ \\ \rightarrow \frac{\sqrt{A+\sqrt{A^{2} -B}} -\sqrt{A-\sqrt{A^{2} -B}}}{\sqrt{2}} \ =P\\ P^{2} =\frac{2A-2\sqrt{\left( A+\sqrt{A^{2} -B}\right)\left( A-\sqrt{A^{2} -B}\right)}}{2}\\ =\frac{2A-2\sqrt{A^{2} -A^{2} +B}}{2} \ =\\ =A-\sqrt{B} \ \\ \rightarrow P=\sqrt{A-\sqrt{B}} \ \\ chứng\ minh\ tương\ tự\ với\ \sqrt{A+\sqrt{B} \ } \ \\ Ta\ được\ dpcm\ \\ áp\ dụng\ \\ Q=\sqrt{7+4\sqrt{3}} \ -\sqrt{7-4\sqrt{3}} \ \\ Q=\sqrt{4+2.2\sqrt{3} +3} -\sqrt{4-2.2\sqrt{3} +3} \ \\ Q=\sqrt{\left( 2+\sqrt{3}\right)^{2}} -\sqrt{\left( 2-\sqrt{3}\right)^{2}} \ \\ Q=2+\sqrt{3} -2+\sqrt{3} =2\sqrt{3} \ \\ P=\sqrt{4+\sqrt{7}} -\sqrt{4-\sqrt{7}}\\ \ P=\frac{\sqrt{8+2\sqrt{7}}}{\sqrt{2}} -\frac{\sqrt{8-2\sqrt{7}}}{\sqrt{2}} \ \\ \rightarrow \frac{1+\sqrt{7} -\sqrt{7} +1}{\sqrt{2}} =\frac{2}{\sqrt{2}} =\sqrt{2} \end{array}$

Q = `\sqrt{7 + 4\sqrt{3}}` -  `\sqrt{7 - 4\sqrt{3}}`

Q = `\sqrt{4 + 2 . 2 . \sqrt{3} + 3}` - `\sqrt{4 - 2 . 2 . \sqrt{3} + 3}`

Q = `\sqrt{(2 + \sqrt{3})²}` - `\sqrt{(2 - \sqrt{3})²}`

Q = 2 + `\sqrt{3}` - 2 + `\sqrt{3}` 

Q = 2`\sqrt{3}`

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P = `\sqrt{4 + \sqrt{7}}` - `\sqrt{4 - \sqrt{7}}`

P = $\frac{\sqrt{8 + 2\sqrt{7}}}{\sqrt{2}}$ - $\frac{\sqrt{8 - 2\sqrt{7}}}{\sqrt{2}}$

P = $\frac{1 + \sqrt{7} -\sqrt{7} + 1}{\sqrt{2}}$

p = $\frac{2}{\sqrt{2}}$ 

P = `\sqrt{2}`