Ta có \(\Delta=m^4-8m-8\)

Để pT có nghiệm nguyên

=> \(\Delta\)là số chính phương, \(\Delta\ge0\)

+ \(m=1\)=> \(\Delta=-15\)loại

+ \(m=2\)=> \(\Delta=-8\)loại

+ \(m=3\)=> \(\Delta=49\)

=> \(x=8;x=1\)nhận

+ m=4 => \(\Delta=216\)loại

+ \(m\ge5\)

=> \(2m^2-8m-9>0\)

=> \(\left(m^2-1\right)^2< m^4-8m-8\)

Mà \(-8m-8< 0\)với \(m\inℤ^+\)

=> \(\left(m^2-1\right)^2< m^4-8m-8< \left(m^2\right)^2\)

Lại có \(m^4-8m-8\)là số chính phương

=> không có giá trị nào của m thỏa mãn

Vậy m=3

 Có \(\left(\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}\right)^2\)

\(=\left(\sqrt{17-3\sqrt{32}}\right)^2+2\left(\sqrt{17-3\sqrt{32}}\right)\left(\sqrt{17+3\sqrt{32}}\right)\)\(+\left(\sqrt{17=3\sqrt{32}}\right)^2\)

 \(=17-3\sqrt{32}+2\sqrt{\left(17-3\sqrt{32}\right)\left(17+3\sqrt{32}\right)}\)\(+17+3\sqrt{32}\)

\(=34+2\sqrt{17^2-9.32}\)

\(=34+2\sqrt{289-288}\)

\(=34+2\sqrt{1}=34+2=36\)

\(\Rightarrow\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}\)

\(=\sqrt{36}=6\)

(Vì có \(\hept{\begin{cases}\sqrt{17-3\sqrt{32}}\ge0\\\sqrt{17+3\sqrt{32}}\ge0\end{cases}}\)nên \(\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}\ge0\))

Ở cuối dòng 2 mình nhầm dấu + thành dấu = nghe mọi người

\(\frac{3}{\sqrt{5}-\sqrt{2}}+\frac{4}{\sqrt{6}+\sqrt{2}}=\frac{3\left(\sqrt{5}+\sqrt{2}\right)}{\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{2}\right)}+\frac{4\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)}\)

\(=\frac{3\left(\sqrt{5}+\sqrt{2}\right)}{3}+\frac{4\left(\sqrt{6}-\sqrt{2}\right)}{4}\)

\(=\sqrt{5}+\sqrt{2}+\sqrt{6}-\sqrt{2}\)

\(=\sqrt{5}+\sqrt{6}\)

ĐKXĐ: Bạn tự làm nha 

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

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

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

\(=\frac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-2\sqrt{x}-1+2\sqrt{x}+2\)

\(=\frac{x^2-\sqrt{x}}{x+\sqrt{x}+1}+1\)

\(=\frac{x^2-\sqrt{x}+x+\sqrt{x}+1}{x+\sqrt{x}+1}\)

\(=\frac{x^2+x+1}{x+\sqrt{x}+1}\)

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

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

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

\(=\frac{\left(\sqrt{a}+1\right)}{\sqrt{a}}.\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\sqrt{a}-1-2}\)

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

\(\left(5\sqrt{7}+7\sqrt{5}\right):\sqrt{35}=\left(\sqrt{5^2.7}+\sqrt{7^2.5}\right):\sqrt{35}\)

\(=\left(\sqrt{35.5}+\sqrt{35.7}\right):\sqrt{35}\)

\(=\sqrt{35}\left(\sqrt{5}+\sqrt{7}\right):\sqrt{35}\)

\(=\sqrt{5}+\sqrt{7}\)

Toán Học Team 

ĐKXĐ: \(x\ge0;x\ne1;\)

\(P=\left(\frac{\sqrt{x}-2}{\sqrt{x}-1}-\frac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right).\frac{\left(1-x\right)^2}{2}\)

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

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

\(=\frac{x\sqrt{x}+2x+\sqrt{x}-2x-4\sqrt{x}-2-x+\sqrt{x}-2\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^1}.\frac{\left(x-1\right)^2}{2}\)

\(=\frac{x\sqrt{x}-x-4\sqrt{x}}{\left(x-1\right)\left(\sqrt{x}+1\right)}.\frac{\left(x-1\right)^2}{2}\)

\(=\frac{\sqrt{x}\left(x-\sqrt{x}-4\right)\left(x-1\right)}{2\left(\sqrt{x}+1\right)}\)

\(=\frac{\sqrt{x}\left(x-\sqrt{x}-4\right)\left(\sqrt{x}-1\right)}{2}\)

Ta có: \(P=\left(\frac{\sqrt{x}-2}{\sqrt{x}-1}-\frac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right)\times\frac{\left(1-x\right)^2}{2}\) 

\(P=\left(\frac{\sqrt{x}-2}{\sqrt{x}-1}-\frac{\sqrt{x}+2}{\left(\sqrt{x}+1\right)^2}\right).\frac{\left(1-x\right)^2}{2}\) 

\(P=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)^2-\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}.\frac{\left(1-x\right)^2}{2}\) 

\(P=\frac{x\sqrt{x}-4\sqrt{x}-x}{-\left(1-x\right)\left(\sqrt{x}+1\right)}.\frac{\left(1-x\right)^2}{2}\) 

\(P=\frac{\sqrt{x}\left(x-4-\sqrt{x}\right)\left(\sqrt{x}-1\right)}{2}\)

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