A=\(\sqrt{48}-3\sqrt{12}+5\sqrt{75}=\sqrt{16.3}-3\sqrt{4.3}+5\sqrt{25.3}=4\sqrt{3}-6\sqrt{3}+25\sqrt{3}=23\sqrt{3}\)
B=\(\dfrac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\dfrac{2-\sqrt{3}}{\sqrt{2}+\sqrt{2-\sqrt{3}}}=\dfrac{\sqrt{2}\left(2+\sqrt{3}\right)}{2+\sqrt{4+2\sqrt{3}}}+\dfrac{\sqrt{2}\left(2-\sqrt{3}\right)}{2+\sqrt{4-2\sqrt{3}}}=\dfrac{2\sqrt{2}+\sqrt{6}}{2+\sqrt{\left(1+\sqrt{3}\right)^2}}+\dfrac{2\sqrt{2}-\sqrt{6}}{2+\sqrt{\left(\sqrt{3}-1\right)^2}}=\dfrac{2\sqrt{2}+\sqrt{6}}{3+\sqrt{3}}+\dfrac{2\sqrt{2}-\sqrt{6}}{\sqrt{3}-1}=\dfrac{\left(2\sqrt{2}+\sqrt{6}\right)\left(\sqrt{3}-1\right)+\left(2\sqrt{2}-\sqrt{6}\right)\left(3+\sqrt{3}\right)}{\sqrt{3}\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}=\dfrac{2\sqrt{6}+3\sqrt{2}-2\sqrt{2}-\sqrt{6}+2\sqrt{6}-3\sqrt{2}+6\sqrt{2}-3\sqrt{6}}{2\sqrt{3}}=\dfrac{4\sqrt{2}}{2\sqrt{3}}=\dfrac{2\sqrt{6}}{3}\)
\(A=\sqrt{16.3}-3\sqrt{4.3}+5\sqrt{25.3}=\sqrt{3}\left(4-6+25\right)=23\sqrt{3}.\)
\(B=\dfrac{\sqrt{2}\left(2+\sqrt{3}\right)}{2+\sqrt{4+2\sqrt{3}}}+\dfrac{\sqrt{2}\left(2-\sqrt{3}\right)}{2+\sqrt{4-2\sqrt{3}}}=\dfrac{2\sqrt{2}+\sqrt{6}}{2+\left(\sqrt{3}+1\right)}+\dfrac{2\sqrt{2}-\sqrt{6}}{2+\sqrt{3}-1}=....\)
b) \(\dfrac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}\) + \(\dfrac{2-\sqrt{3}}{\sqrt{2}+\sqrt{2-\sqrt{3}}}\)
= \(\dfrac{2\sqrt{2}+\sqrt{6}}{2+\sqrt{4+2\sqrt{3}}}\) + \(\dfrac{2\sqrt{2}-\sqrt{6}}{2+\sqrt{4-2\sqrt{3}}}\)
= \(\dfrac{2\sqrt{2}+\sqrt{6}}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}\) + \(\dfrac{2\sqrt{2}-\sqrt{6}}{2+\sqrt{\left(\sqrt{3}-1\right)^2}}\)
= \(\dfrac{2\sqrt{2}+\sqrt{6}}{3+\sqrt{3}}\) + \(\dfrac{2\sqrt{2}-\sqrt{6}}{\sqrt{3}+1}\)
= \(\dfrac{2\sqrt{2}+\sqrt{6}}{\sqrt{3}\left(\sqrt{3}+1\right)}\) + \(\dfrac{2\sqrt{2}-\sqrt{6}}{\sqrt{3}+1}\)
= \(\dfrac{2\sqrt{2}+\sqrt{6}+\left(2\sqrt{2}-\sqrt{6}\right)\sqrt{3}}{\sqrt{3}\left(\sqrt{3}+1\right)}\)
= \(\dfrac{2\sqrt{2}+\sqrt{6}+2\sqrt{6}-3\sqrt{2}}{\sqrt{3}\left(\sqrt{3}+1\right)}\)
= \(\dfrac{3\sqrt{6}-\sqrt{2}}{\sqrt{3}\left(\sqrt{3}+1\right)}\)
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