Question

Chứng minh rằng sqrt(10 + 2sqrt(24)) - sqrt(10 - 2sqrt(24)) = 4

Answer 1

\(\sqrt{10+2\sqrt{24}}-\sqrt{10-2\sqrt{24}}\)

\(=\sqrt{6+2\cdot\sqrt{6}\cdot2+4}-\sqrt{6-2\cdot\sqrt{6}\cdot2+4}\)

\(=\sqrt{\left(\sqrt{6}+2\right)^2}-\sqrt{\left(\sqrt{6}-2\right)^2}\)

\(=\left(\sqrt{6}+2\right)-\left(\sqrt{6}-2\right)\)

\(=\sqrt{6}+2-\sqrt{6}+2=4\)

Answer 2

Ta có: VT = \(\sqrt{10+2\sqrt{24}}-\sqrt{10-2\sqrt{24}}\)

\(=\sqrt{6+2\cdot2\cdot\sqrt{6}+4}-\sqrt{6-2\cdot2\cdot\sqrt{6}+4}\)

\(=\sqrt{\left(\sqrt{6}+2\right)^2}-\sqrt{\left(\sqrt{6}-2\right)^2}\)

\(=\sqrt{6}+2-\sqrt{6}+2=4\)

VT= VP \(\Rightarrow\) đpcm