\(a,\left(3-\frac{\sqrt{5}\left(\sqrt{5}+\sqrt{3}\right)}{\sqrt{5}+\sqrt{3}}\right)\left(3+\frac{\sqrt{5}\left(\sqrt{5}-1\right)}{\sqrt{5}-1}\right)\)

\(\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)\)

\(3^2-\sqrt{5}^2\)

\(=4\)

\(b,\frac{6\sqrt{2}-2\sqrt{6}}{\sqrt{6}-\sqrt{2}}+\frac{3}{3-\sqrt{2}}\)

\(\frac{\sqrt{6}\sqrt{2}\left(\sqrt{6}-\sqrt{2}\right)}{\sqrt{6}-\sqrt{2}}+\frac{3}{3-\sqrt{2}}\)

\(\sqrt{12}+\frac{3}{3-\sqrt{2}}\)

\(\frac{6\sqrt{3}-2\sqrt{6}-3}{3-\sqrt{2}}\)

\(c,\frac{4+2\sqrt{3}}{4-\sqrt{15}}\frac{\left(\sqrt{5}-\sqrt{3}\right)}{2}-2\sqrt{3}\)

\(\frac{2+\sqrt{3}\left(\sqrt{5}-\sqrt{3}\right)}{4-\sqrt{15}}-2\sqrt{3}\)

\(\frac{2\sqrt{5}+\sqrt{15}-2\sqrt{3}-3}{4-\sqrt{15}}-2\sqrt{3}\)

\(\frac{2\sqrt{5}+\sqrt{15}-2\sqrt{3}-3+6\sqrt{5}-8\sqrt{3}}{4-\sqrt{15}}\)

\(\frac{8\sqrt{5}+\sqrt{15}-10\sqrt{3}-3}{4-\sqrt{15}}\)

\(\sqrt{49a^2}+3a\)

\(\sqrt{49}\sqrt{a^2}+3a\)

\(7a+3a\)

\(=10a\)

Đk: \(a^2>b^2\); b \(\ne\)0

Q = \(\frac{a}{\sqrt{a^2-b^2}}-\left(1+\frac{a}{\sqrt{a^2-b^2}}\right):\frac{b}{a-\sqrt{a^2-b^2}}\)

Q = \(\frac{a}{\sqrt{a^2-b^2}}-\frac{\sqrt{a^2-b^2}+a}{\sqrt{a^2-b^2}}\cdot\frac{a-\sqrt{a^2-b^2}}{b}\)

Q = \(\frac{a}{\sqrt{a^2-b^2}}-\frac{a^2-\left(a^2-b^2\right)}{b\sqrt{a^2-b^2}}\)

Q = \(\frac{ab-b^2}{b\sqrt{a^2-b^2}}=\frac{b\left(a-b\right)}{b\sqrt{\left(a-b\right)\left(a+b\right)}}=\frac{\sqrt{a-b}}{\sqrt{a+b}}\)

\(abc\le\left(\frac{a+b+c}{3}\right)^3\Leftrightarrow\sqrt[3]{abc}\le\frac{a+b+c}{3}\)

BĐT Cô- si 

đánh giá từ tbn sang tbc đấy bạn 

\(A=\frac{2x-3\sqrt{x}-2}{\sqrt{x}-2}\)

\(A=\frac{2x+\sqrt{x}-4\sqrt{x}-2}{\sqrt{x}-2}\)

\(A=\frac{\sqrt{x}\left(2\sqrt{x}+1\right)-2\left(2\sqrt{x}+1\right)}{\sqrt{x}-2}\)

\(A=\frac{\sqrt{x}-2\left(2\sqrt{x}+1\right)}{\sqrt{x}-2}\)

\(A=2\sqrt{x}+1\)

\(\frac{1}{2+\sqrt{5}+2\sqrt{2}+\sqrt{10}}=\frac{1}{2+\sqrt{5}+\sqrt{2}\left(2+\sqrt{5}\right)}=\frac{1}{\left(\sqrt{2}+1\right)\left(2+\sqrt{5}\right)}\)

\(=\frac{\left(\sqrt{2}-1\right)\left(\sqrt{5}-2\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)}=\sqrt{10}-2\sqrt{2}-\sqrt{5}+2\)

\(\left(\frac{x+\sqrt{x}-1}{x\sqrt{x}-1}-\frac{\sqrt{x}+1}{x+\sqrt{x}+1}\right):\frac{1}{\sqrt{x}-1}\)

\(\frac{x+\sqrt{x}-1-\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\sqrt{x}-1\)

\(\frac{x+\sqrt{x}-1-x-\sqrt{x}+\sqrt{x}+1}{x+\sqrt{x}+1}\)

\(\frac{\sqrt{x}}{x+\sqrt{x}+1}\)

\(=\left(\frac{x+\sqrt{x}-1}{\left(\sqrt{x}\right)^3-1}-\frac{\sqrt{x}+1}{x+\sqrt{x}+1}\right):\frac{1}{\sqrt{x}-1}\)

\(=\left(\frac{x+\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right):\frac{1}{\sqrt{x}-1}\)

\(=\frac{x+\sqrt{x}-1-x+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\frac{\sqrt{x}-1}{1}\)

\(=\frac{\sqrt{x}}{x+\sqrt{x}+1}\)