\(a,=\dfrac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\dfrac{x-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\dfrac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{x+2+x-1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\dfrac{x-\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\dfrac{\sqrt{x}}{x+\sqrt{x}+1}\)
Với \(x=28-6\sqrt{3}tmđk\)thay vào P ta có :
\(P=\dfrac{\sqrt{28-6\sqrt{3}}}{28-6\sqrt{3}+\sqrt{28-6\sqrt{3}}+1}=\dfrac{\sqrt{\left(3\sqrt{3}-1\right)^2}}{29-6\sqrt{3}+\sqrt{\left(3\sqrt{3}-1\right)^2}}=\dfrac{3\sqrt{3}-1}{29-6\sqrt{3}+3\sqrt{3}-1}=\dfrac{3\sqrt{3}-1}{28-3\sqrt{3}}=\dfrac{\left(3\sqrt{3}-1\right)\left(28+3\sqrt{3}\right)}{784-27}=\dfrac{81\sqrt{3}-1}{757}\)
\(=\left[\dfrac{1+\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}-\dfrac{1-\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}\right]\cdot\dfrac{1-\sqrt{a}}{\sqrt{a}}\)
\(=\dfrac{2\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}\cdot\dfrac{1-\sqrt{a}}{\sqrt{a}}=\dfrac{2}{\sqrt{a}+1}\)
Với \(a=3-2\sqrt{2}tmđk\)thay vào M ta được :
\(M=\dfrac{2}{\sqrt{3-2\sqrt{2}}+1}=\dfrac{2}{\sqrt{\left(\sqrt{2}-1\right)^2}+1}=\dfrac{2}{\sqrt{2}-1+1}=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)
Ta có : \(18M=18\cdot\dfrac{2}{\sqrt{a}+1}=\dfrac{36}{\sqrt{a}+1}\)
Đặt \(\dfrac{36}{\sqrt{a}+1}=x^2\left(x\in N\cdot\right)\Rightarrow x^2\left(\sqrt{a}+1\right)=36\)
Ta lại có a2.b2 = (a.b)2 => \(\left\{{}\begin{matrix}x^2\\\sqrt{a}+1\end{matrix}\right.\)phải là bình phương của các số tự nhiên
mà \(x^2\left(\sqrt{a}+1\right)=36\)=> Ta có các trường hợp sau :
\(\left\{{}\begin{matrix}x^2=1\\\sqrt{a}+1=36\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\a=1225\end{matrix}\right.\)(tm)
\(\left\{{}\begin{matrix}x^2=36\\\sqrt{a}+1=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=6\\a=0\end{matrix}\right.\)(ktm)
\(\left\{{}\begin{matrix}x^2=4\\\sqrt{a}+1=9\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2\\a=64\end{matrix}\right.\)(tm)
\(\left\{{}\begin{matrix}x^2=9\\\sqrt{a}+1=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=3\\a=9\end{matrix}\right.\)(tm)
Bài 12:
a: Ta có: \(P=\dfrac{x+2}{x\sqrt{x}+1}+\dfrac{\sqrt{x}+1}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\)
\(=\dfrac{x+2+x-1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{x-\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{\sqrt{x}}{x+\sqrt{x}+1}\)
b: Thay \(x=28-6\sqrt{3}\) vào P, ta được:
\(P=\dfrac{3\sqrt{3}-1}{28-6\sqrt{3}+3\sqrt{3}-1+1}=\dfrac{3\sqrt{3}-1}{28-3\sqrt{3}}=\dfrac{81\sqrt{3}-1}{757}\)
Bài 13:
a: Ta có: \(M=\left(\dfrac{1}{1-\sqrt{a}}-\dfrac{1}{1+\sqrt{a}}\right)\cdot\left(\dfrac{1}{\sqrt{a}}-1\right)\)
\(=\dfrac{1+\sqrt{a}-1+\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}\cdot\dfrac{1-\sqrt{a}}{\sqrt{a}}\)
\(=\dfrac{2}{\sqrt{a}+1}\)
b: Thay \(a=3-2\sqrt{2}\) vào M, ta được:
\(M=\dfrac{2}{\sqrt{2}-1+1}=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)
