chứng minh các đẳng thức sau:
a) \(\sqrt{\dfrac{2-\sqrt{3}}{2+\sqrt{3}}}\) + \(\sqrt{\dfrac{2+\sqrt{3}}{2-\sqrt{3}}}\) = 4
b) \(\dfrac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}\) - \(\dfrac{\sqrt{b}}{\sqrt{a}+\sqrt{b}}\) - \(\dfrac{2b}{a-b}\) = 1 với ≥ 0, b ≥ 0, a ≠ b;
c) \(\left(1+\dfrac{a+\sqrt{a}}{\sqrt{a}+1}\right)\)\(\left(1-\dfrac{a-\sqrt{a}}{\sqrt{a}-1}\right)\) = 1 - a với a > 0, a ≠ 1
b) \(\dfrac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}-\dfrac{\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\dfrac{2b}{a-b}\)
\(=\dfrac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}-\dfrac{\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\dfrac{2b}{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}\)
\(=\dfrac{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)-\sqrt{b}\left(\sqrt{a}-\sqrt{b}\right)-2b}{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}\)
\(=\dfrac{a+\sqrt{ab}-\sqrt{ab}+b-\sqrt{ab}+b-2b}{a-b}\)
\(=\dfrac{a}{a-b}\)
khúc \(\dfrac{a}{a-b}\) sai nhé
\(=\dfrac{a-b}{a-b}=1\)
Câu a : \(VT=\sqrt{\dfrac{2-\sqrt{3}}{2+\sqrt{3}}}+\sqrt{\dfrac{2+\sqrt{3}}{2-\sqrt{3}}}\)
\(=\sqrt{\dfrac{2\left(2-\sqrt{3}\right)}{2\left(2+\sqrt{3}\right)}}+\sqrt{\dfrac{2\left(2+\sqrt{3}\right)}{2\left(2-\sqrt{3}\right)}}\)
\(=\sqrt{\dfrac{4-2\sqrt{3}}{4+2\sqrt{3}}}+\sqrt{\dfrac{4+2\sqrt{3}}{4-2\sqrt{3}}}\)
\(=\sqrt{\dfrac{3-2\sqrt{3}+1}{3+2\sqrt{3}+1}}+\sqrt{\dfrac{3+2\sqrt{3}+1}{3-2\sqrt{3}+1}}\)
\(=\sqrt{\dfrac{\left(\sqrt{3}-1\right)^2}{\left(\sqrt{3}+1\right)^2}}+\sqrt{\dfrac{\left(\sqrt{3}+1\right)^2}{\left(\sqrt{3}-1\right)^2}}\)
\(=\dfrac{\sqrt{3}-1}{\sqrt{3}+1}+\dfrac{\sqrt{3}+1}{\sqrt{3}-1}\)
\(=\dfrac{\left(\sqrt{3}-1\right)^2+\left(\sqrt{3}+1\right)^2}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}\)
\(=\dfrac{3-2\sqrt{3}+1+3+2\sqrt{3}+1}{3-1}\)
\(=\dfrac{8}{2}=4\) ( đpcm )
Câu c : \(VT=\left(1+\dfrac{a+\sqrt{a}}{\sqrt{a}+1}\right)\left(1-\dfrac{a-\sqrt{a}}{\sqrt{a}-1}\right)\)
\(=\left(1+\dfrac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}\right)\left(1-\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right)\)
\(=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)=1-a\) ( đpcm )
