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a. \(=\dfrac{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}+\sqrt{b}}=\sqrt{a}\)
b. \(=\dfrac{1-\left(2\sqrt{a}\right)^3}{1-2\sqrt{a}}=\dfrac{\left(1-2\sqrt{a}\right)\left(1+2\sqrt{a}+4a\right)}{1-2\sqrt{a}}=1+2\sqrt{a}+4a\)
c. \(=\dfrac{1-\left(\sqrt{a}\right)^2}{1+\sqrt{a}}=\dfrac{\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)}{1+\sqrt{a}}=1-\sqrt{a}\)
d. \(=\dfrac{\sqrt{a}\left(\sqrt{a}-3\right)}{\sqrt{a}-3}=\sqrt{a}\)
\(x=\sqrt[3]{a+\dfrac{a+1}{3}\sqrt{\dfrac{8a-1}{3}}}+\sqrt[3]{a-\dfrac{a+1}{3}\sqrt{\dfrac{8a-1}{3}}}\\ >\sqrt[3]{\dfrac{1}{8}+\dfrac{a+1}{3}\sqrt{\dfrac{8\cdot\dfrac{1}{8}-1}{3}}}+\sqrt[3]{\dfrac{1}{8}-\dfrac{a+1}{3}\sqrt{\dfrac{8\cdot\dfrac{1}{8}-1}{3}}}\\ =\sqrt[3]{\dfrac{1}{8}+\dfrac{a+1}{3}\sqrt{\dfrac{1-1}{3}}}+\sqrt[3]{\dfrac{1}{8}-\dfrac{a+1}{3}\sqrt{\dfrac{1-1}{3}}}\\ =\sqrt[3]{\dfrac{1}{8}}+\sqrt[3]{\dfrac{1}{8}}=\dfrac{1}{2}+\dfrac{1}{2}=1>0\)
Vậy................
Xét \(x^3=2a+3x.\sqrt[3]{a^2-\left(\dfrac{a+1}{3}\right)^2.\dfrac{8a-1}{3}}\)
\(\Leftrightarrow x^3=2a+3x.\sqrt[3]{\dfrac{\left(1-2a\right)^3}{27}}\)
\(\Leftrightarrow x^3=2a+x.\left(1-2a\right)\)
\(\Leftrightarrow\left(x-1\right)\left(x^2+x+2a\right)=0\)
Dễ thấy \(x^2+x+2a=\left(x+\dfrac{1}{2}\right)^2+\dfrac{8a-1}{4}>0\) (vì \(a>\dfrac{1}{8}\))
Nên x=1 hay x là số nguyên.
a) điều kiện xác định : \(a\ge0;a\ne1\)
ta có : \(P=\dfrac{3a+\sqrt{9a}-3}{a+\sqrt{a}-2}-\dfrac{\sqrt{a}-2}{\sqrt{a}-1}+\dfrac{1}{\sqrt{a}+2}-1\)
\(\Leftrightarrow P=\dfrac{3a+3\sqrt{a}-3}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+2\right)}-\dfrac{\sqrt{a}-2}{\sqrt{a}-1}-\dfrac{\sqrt{a}+1}{\sqrt{a}+2}\) \(\Leftrightarrow P=\dfrac{3a+3\sqrt{a}-3-\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)-\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+2\right)}\) \(\Leftrightarrow P=\dfrac{a+3\sqrt{a}+2}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+2\right)}=\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{a}+2\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+2\right)}=\dfrac{\sqrt{a}+1}{\sqrt{a}-1}\)để \(\left|P\right|=1\Leftrightarrow\left|\dfrac{\sqrt{a}+1}{\sqrt{a}-1}\right|=1\) \(\Leftrightarrow\left[{}\begin{matrix}\dfrac{\sqrt{a}+1}{\sqrt{a}-1}=1\\\dfrac{\sqrt{a}+1}{\sqrt{a}-1}=-1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{\sqrt{a}+1}{\sqrt{a}-1}-1=0\\\dfrac{\sqrt{a}+1}{\sqrt{a}-1}+1=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\dfrac{2}{\sqrt{a}-1}=0\\\dfrac{2\sqrt{a}}{\sqrt{a}-1}=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}2=0\left(vôlí\right)\\2\sqrt{a}=0\end{matrix}\right.\Rightarrow a=0\)
vậy \(a=0\)
Đặt biểu thức trên là A
\(A^3=2a+3A\sqrt[3]{\left(a+\dfrac{a+1}{3}\sqrt{\dfrac{8a-1}{3}}\right)\left(a-\dfrac{a+1}{3}\sqrt{\dfrac{8a-1}{3}}\right)}\)
\(=2a+3A\sqrt[3]{a^2-\left(\dfrac{a+1}{3}\right)^2.\dfrac{8a-1}{3}}\)
\(=2a+3A\sqrt[3]{\dfrac{-8a^3+12a^2-6a+1}{27}}\)
\(=2a+3A\sqrt[3]{\left(\dfrac{1-2a}{3}\right)^3}=2a+A\left(1-2a\right)\)
\(\Leftrightarrow A^3-2a-A+2aA=0\)
\(\Leftrightarrow\left(A-1\right)\left(A^2+A+2a\right)=0\)
Dễ thấy \(A^2+A+2a>0\) nên A=1.
\(A=\dfrac{2-\sqrt{a}-\sqrt{a}-3}{2\sqrt{a}+1}=-1\)
\(B=\dfrac{1}{1-\sqrt{2+\sqrt{3}}}-\dfrac{1}{1-\sqrt{2-\sqrt{3}}}\)
\(=\dfrac{\sqrt{2}}{\sqrt{2}-\sqrt{3}-1}-\dfrac{\sqrt{2}}{\sqrt{2}-\sqrt{3}+1}\)
\(=\dfrac{2-\sqrt{6}+\sqrt{2}-2+\sqrt{6}+\sqrt{2}}{5-2\sqrt{6}-1}\)
\(=\dfrac{2\sqrt{2}}{4-2\sqrt{6}}=\dfrac{1}{\sqrt{2}-\sqrt{3}}=-\sqrt{2}-\sqrt{3}\)