Question

CMR:

\(\sqrt{1+\dfrac{1}{2}\sqrt{3}}-\sqrt{1-\dfrac{1}{2}\sqrt{3}}=1\)

Answer 1

\(\sqrt{1+\dfrac{1}{2}\sqrt{3}}-\sqrt{1-\dfrac{1}{2}\sqrt{3}}\)

\(=\sqrt{\dfrac{4+2\sqrt{3}}{4}}-\sqrt{\dfrac{4-2\sqrt{3}}{4}}\)

\(=\dfrac{1}{2}\left(\sqrt{\left(\sqrt{3}+1\right)^2}\right)-\sqrt{\left(\sqrt{3}-1\right)^2}\)

\(=\dfrac{1}{2}\left(\left|\sqrt{3}+1\right|-\left|\sqrt{3}-1\right|\right)\)

\(=\dfrac{1}{2}\left(\sqrt{3}+1-\sqrt{3}+1\right)=1\left(\text{đ}pcm\right)\)