\(\frac{9+4\sqrt{2}}{21}\)

cho  P = \(\frac{\sqrt{x}+2}{\sqrt{x}+1}\)  , Tìm GTLN của P  

ĐKXĐ: \(x\ge0\)

Ta có: \(P=\frac{\sqrt{x}+2}{\sqrt{x}+1}=1+\frac{1}{\sqrt{x}+1}\)

Để P lớn nhất thì: \(\frac{1}{\sqrt{x}+1}\)phải lớn nhất.Hay: \(\sqrt{x}+1\)nhỏ nhất

Theo ĐKXĐ,lại có: \(x\ge0\Rightarrow\sqrt{x}\ge0\Rightarrow\sqrt{x}+1\ge1\)

=>Min \(\sqrt{x}+1\)là 1 tại \(\sqrt{x}=0\Rightarrow x=0\)

=>Max P = \(1+\frac{1}{0+1}=2\)tại x=0

=.= hk tốt!!

\(C=\dfrac{\sqrt{10}-\sqrt{5}+2\sqrt{2}+\sqrt{5}-\sqrt{10}-1}{2\sqrt{2}+2+2\sqrt{2}-1+2\sqrt{2}+2}\)

\(=\dfrac{2\sqrt{2}-1}{6\sqrt{2}+3}=\dfrac{9-4\sqrt{2}}{21}\)

\(B=\dfrac{40}{6+2\sqrt{5}+\sqrt{4\sqrt{5}+4}}\)

\(=\dfrac{40}{\left(\sqrt{5}+1\right)^2+2\sqrt{\sqrt{5}+1}}\)

\(=\dfrac{40}{\sqrt{\sqrt{5}+1}\left(\sqrt{\sqrt{5}+1}+2\right)}\)

\(=\dfrac{40\sqrt{\sqrt{5}-1}}{2\left(\sqrt{\sqrt{5}+1}+2\right)}\)

\(=\dfrac{20\left(\sqrt{\sqrt{5}-1}\right)\left(\sqrt{\sqrt{5}+1}-2\right)}{\sqrt{5}+1-4}\)

\(=\dfrac{20\left(\sqrt{\sqrt{5}-1}\right)\left(\sqrt{\sqrt{5}+1}-2\right)}{-3+\sqrt{5}}\)

\(=-5\left(3+\sqrt{5}\right)\left(\sqrt{\sqrt{5}-1}\right)\left(\sqrt{\sqrt{5}+1}-2\right)\)

a: \(A=\left(4+\sqrt{15}\right)\cdot\left(\sqrt{5}-\sqrt{3}\right)\cdot\sqrt{8-2\sqrt{15}}\)

\(=\left(4+\sqrt{15}\right)\left(8-2\sqrt{15}\right)\)

\(=32-8\sqrt{15}+8\sqrt{15}-30=2\)

b: \(\sqrt{2}\cdot B=\left(3-\sqrt{5}\right)\left(\sqrt{5}+1\right)+\left(3+\sqrt{5}\right)\left(\sqrt{5}-1\right)\)

\(\Leftrightarrow B\sqrt{2}=3\sqrt{5}+3-5-\sqrt{5}+3\sqrt{5}-3+5-\sqrt{5}\)

\(\Leftrightarrow B\sqrt{2}=4\sqrt{5}\)

hay \(B=2\sqrt{10}\)

d: \(D\sqrt{2}=\sqrt{5}+\sqrt{3}+\sqrt{5}-\sqrt{3}-2\cdot\left(\sqrt{5}-1\right)\)

\(=2\sqrt{5}-2\sqrt{5}+2=2\)

hay \(D=\sqrt{2}\)

 kho wa do

a) Ta có: \(\left(7\sqrt{48}+3\sqrt{27}-2\sqrt{12}\right)\cdot\sqrt{3}\)

\(=\left(7\cdot4\sqrt{3}+3\cdot3\sqrt{3}-2\cdot2\sqrt{3}\right)\cdot\sqrt{3}\)

\(=33\sqrt{3}\cdot\sqrt{3}\)

=99

b) Ta có: \(\left(12\sqrt{50}-8\sqrt{200}+7\sqrt{450}\right):\sqrt{10}\)

\(=\left(12\cdot5\sqrt{2}-8\cdot10\sqrt{2}+7\cdot15\sqrt{2}\right):\sqrt{10}\)

\(=\dfrac{85\sqrt{2}}{\sqrt{10}}=\dfrac{85}{\sqrt{5}}=17\sqrt{5}\)

c) Ta có: \(\left(2\sqrt{6}-4\sqrt{3}+5\sqrt{2}-\dfrac{1}{4}\sqrt{8}\right)\cdot3\sqrt{6}\)

\(=\left(2\sqrt{6}-4\sqrt{3}+5\sqrt{2}-\dfrac{1}{4}\cdot2\sqrt{2}\right)\cdot3\sqrt{6}\)

\(=\left(2\sqrt{6}-4\sqrt{3}+3\sqrt{2}\right)\cdot3\sqrt{6}\)

\(=36-36\sqrt{2}+18\sqrt{3}\)

d) Ta có: \(3\sqrt{15\sqrt{50}}+5\sqrt{24\sqrt{8}}-4\sqrt{12\sqrt{32}}\)

\(=3\cdot\sqrt{75\sqrt{2}}+5\cdot\sqrt{48\sqrt{2}}-4\sqrt{48\sqrt{2}}\)

\(=3\cdot5\sqrt{2}\cdot\sqrt{\sqrt{2}}+4\sqrt{3}\sqrt{\sqrt{2}}\)

\(=15\sqrt{\sqrt{8}}+4\sqrt{\sqrt{18}}\)

a,=\(\left(28\sqrt{3}+9\sqrt{3}-4\sqrt{3}\right).\sqrt{3}\)

   \(=28.3+9.3-4.3=99\)

b,\(=\left(60\sqrt{2}-80\sqrt{2}+175\sqrt{2}\right):\sqrt{10}\)

  \(=155\sqrt{2}:\sqrt{10}=\dfrac{155}{\sqrt{5}}\)

d,Ta có:\(3\sqrt{15\sqrt{50}}+5\sqrt{24\sqrt{8}}-4\sqrt{12\sqrt{32}}\)

        \(=3\sqrt{75\sqrt{2}}+5\sqrt{48\sqrt{2}}-4\sqrt{48\sqrt{2}}\)

        \(=15\sqrt{3\sqrt{2}}+20\sqrt{3\sqrt{2}}-16\sqrt{3\sqrt{2}}\)

        \(=19\sqrt{3\sqrt{2}}\)