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\(=\frac{\left(\sqrt{a}-\sqrt{b}\right)^3+2\sqrt{a^3}+\sqrt{b^3}}{3\sqrt{a}\left(\sqrt{a^3}+\sqrt{b^3}\right)}+\frac{\sqrt{a}\left(\sqrt{b}-\sqrt{a}\right)}{\sqrt{a}\left(a-b\right)}\)
\(=\frac{\sqrt{a^3}-3a\sqrt{b}+3\sqrt{a}.b-\sqrt{b^3}+2\sqrt{a^3}+\sqrt{b^3}}{3\sqrt{a}\left(\sqrt{a^3}+\sqrt{b^3}\right)}+\frac{\sqrt{a}\left(\sqrt{b}-\sqrt{a}\right)}{\sqrt{a}\left(a-b\right)}\)
\(=\frac{3\sqrt{a^3}-3a\sqrt{b}+3b\sqrt{a}}{3\sqrt{a}\left(\sqrt{a^3}+\sqrt{b^3}\right)}+\frac{\sqrt{a}\left(\sqrt{b}-\sqrt{a}\right)}{\sqrt{a}\left(a-b\right)}\)
\(=\frac{a-\sqrt{ab}+b}{\left(\sqrt{a}+\sqrt{b}\right)\left(a-\sqrt{ab}+b\right)}-\frac{1}{\sqrt{a}+\sqrt{b}}=0\)
1. Ta thấy:
\(\frac{(a-b)^3}{(\sqrt{a}-\sqrt{b})^3}-b\sqrt{b}+2a\sqrt{a}=\frac{(\sqrt{a}-\sqrt{b})^3(\sqrt{a}+\sqrt{b})^3}{(\sqrt{a}-\sqrt{b})^3}-b\sqrt{b}+2a\sqrt{a}\)
\(=(\sqrt{a}+\sqrt{b})^3-b\sqrt{b}+2a\sqrt{a}=a\sqrt{a}+b\sqrt{b}+3\sqrt{ab}(\sqrt{a}+\sqrt{b})-b\sqrt{b}+2a\sqrt{a}\)
\(=3a\sqrt{a}+3\sqrt{ab}(\sqrt{a}+\sqrt{b})=3\sqrt{a}(a+\sqrt{ab}+b)\)
$a\sqrt{a}-b\sqrt{b}=(\sqrt{a}-\sqrt{b})(a+\sqrt{ab}+b)$
\(\frac{\frac{(a-b)^3}{(\sqrt{a}-\sqrt{b})^3}-b\sqrt{b}+2a\sqrt{a}}{a\sqrt{a}-b\sqrt{b}}=\frac{3\sqrt{a}}{\sqrt{a}-\sqrt{b}}(1)\)
\(\frac{3a+3\sqrt{ab}}{b-a}=\frac{3\sqrt{a}(\sqrt{a}+\sqrt{b})}{(\sqrt{b}-\sqrt{a})(\sqrt{b}+\sqrt{a})}=\frac{-3\sqrt{a}}{\sqrt{a}-\sqrt{b}}(2)\)
Từ $(1);(2)$ ta có đpcm.
Câu 2:
Điều kiện đã cho tương đương với:
$\frac{a-b}{a(a+b)}+\frac{a+b}{a(a-b)}=\frac{3a-b}{(a-b)(a+b)}$
$\Leftrightarrow \frac{(a-b)^2}{a(a+b)(a-b)}+\frac{(a+b)^2}{a(a-b)(a+b)}=\frac{a(3a-b)}{a(a-b)(a+b)}$
$\Leftrightarrow (a-b)^2+(a+b)^2=a(3a-b)$
$\Leftrightarrow 2a^2+2b^2=3a^2-ab$
$\Leftrightarrow a^2-ab-2b^2=0$
$\Leftrightarrow (a+b)(a-2b)=0$
$\Leftrightarrow a=-b$ hoặc $a=2b$
Nếu $a=-b$ thì $|a|=|b|$ (trái giả thiết). Do đó $a=2b$
Khi đó:
$P=\frac{(2b)^3+2(2b)^2.b+3b^3}{2(2b)^3+2b.b^2+b^3}=\frac{19b^3}{19b^3}=1$
\(S=\frac{\left[\frac{\left(a-b\right)\left(\sqrt{a}-\sqrt{b}\right)}{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)}\right]^3+2a\sqrt{a}+b\sqrt{b}}{3a^2+3b\sqrt{ab}}+\frac{\sqrt{a}\left(\sqrt{b}-\sqrt{a}\right)}{\sqrt{a}\left(a-b\right)}\)
\(S=\frac{\left(\sqrt{a}-\sqrt{b}\right)^3+2\left(\sqrt{a}\right)^2\sqrt{a}+\left(\sqrt{b}\right)^2\sqrt{b}}{3a^2+3b\sqrt{ab}}+\frac{\sqrt{b}-\sqrt{a}}{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}\)
\(S=\frac{\left(\sqrt{a}\right)^3-3\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)-\left(\sqrt{b}\right)^3+2\left(\sqrt{a}\right)^3+\left(\sqrt{b}\right)^3}{3a^2+3b\sqrt{ab}}-\frac{1}{\sqrt{a}+\sqrt{b}}\)
\(S=\frac{3\left(\sqrt{a}\right)^3-3a\sqrt{b}+3\sqrt{a}b}{3a^2+3b\sqrt{ab}}-\frac{1}{\sqrt{a}+\sqrt{b}}\)
\(S=\frac{\sqrt{a}\left(a-\sqrt{ab}+b\right)}{\sqrt{a}\left[\left(\sqrt{a}\right)^3+\left(\sqrt{b}\right)^3\right]}-\frac{1}{\sqrt{a}+\sqrt{b}}\)
\(S=\frac{a-\sqrt{ab}+b}{\left(\sqrt{a}+\sqrt{b}\right)\left(a-\sqrt{ab}+b\right)}-\frac{1}{\sqrt{a}+\sqrt{b}}\)
\(S=\frac{1}{\sqrt{a}+\sqrt{b}}-\frac{1}{\sqrt{a}+\sqrt{b}}=0\)
1. ĐK \(\hept{\begin{cases}x\ge0\\x\ne4\end{cases}}\)
a. Ta có \(R=\left(\frac{\sqrt{x}}{\sqrt{x}-2}-\frac{4}{\sqrt{x}\left(\sqrt{x}-2\right)}\right).\left(\frac{1}{\sqrt{x}+2}+\frac{4}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\right)\)
\(=\frac{x-4}{\sqrt{x}\left(\sqrt{x}-2\right)}.\frac{\sqrt{x}-2+4}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\frac{\sqrt{x}+2}{\sqrt{x}}.\frac{\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
\(=\frac{\sqrt{x}+2}{\sqrt{x}\left(\sqrt{x}-2\right)}\)
b. Với \(x=4+2\sqrt{3}\Rightarrow R=\frac{\sqrt{4+2\sqrt{3}}+2}{\sqrt{4+2\sqrt{3}}\left(\sqrt{4+2\sqrt{3}}-2\right)}=\frac{\sqrt{\left(\sqrt{3}+1\right)^2}+2}{\sqrt{\left(\sqrt{3}+1\right)^2}\left(\sqrt{\left(\sqrt{3}+1\right)^2}-2\right)}\)
\(=\frac{\sqrt{3}+1+2}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}=\frac{\sqrt{3}+3}{3-1}=\frac{\sqrt{3}+3}{2}\)
c. Để \(R>0\Rightarrow\frac{\sqrt{x}+2}{\sqrt{x}\left(\sqrt{x}-2\right)}>0\Rightarrow\sqrt{x}-2>0\Rightarrow x>4\)
Vậy \(x>4\)thì \(R>0\)
2. Ta có \(A=6+2\sqrt{2}=6+\sqrt{8};B=9=6+3=6+\sqrt{9}\)
Vì \(\sqrt{8}< \sqrt{9}\Rightarrow A< B\)
3. a. \(VT=\frac{a+b-2\sqrt{ab}}{\sqrt{a}-\sqrt{b}}:\frac{1}{\sqrt{a}+\sqrt{b}}=\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{\sqrt{a}-\sqrt{b}}.\left(\sqrt{a}+\sqrt{b}\right)\)
\(=\left(\sqrt{a}-\sqrt{b}\right).\left(\sqrt{a}+\sqrt{b}\right)=a-b=VP\left(đpcm\right)\)
b. Ta có \(VT=\left(2+\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right).\left(2-\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}\right)\)
\(=\left(2+\sqrt{a}\right)\left(2-\sqrt{a}\right)=4-a=VP\left(đpcm\right)\)
\( Q = \dfrac{{{{\left( {\dfrac{{a - b}}{{\sqrt a + \sqrt b }}} \right)}^3} + 2a\sqrt a + b\sqrt b }}{{3{a^2} + 3b\sqrt {ab} }} + \dfrac{{\sqrt {ab} - a}}{{a\sqrt a - b\sqrt a }}\\ Q = \dfrac{{{{\left[ {\dfrac{{\left( {\sqrt a - \sqrt b } \right)\left( {\sqrt a + \sqrt b } \right)}}{{\sqrt a + \sqrt b }}} \right]}^3} + 2a\sqrt a + b\sqrt b }}{{3\left( {{a^2} + b\sqrt {ab} } \right)}} + \dfrac{{\sqrt a \left( {\sqrt b - \sqrt a } \right)}}{{\sqrt a \left( {a - b} \right)}}\\ Q = \dfrac{{{{\left( {\sqrt a - \sqrt b } \right)}^3} + 2a\sqrt a + b\sqrt b }}{{3\sqrt a \left( {a\sqrt a + b\sqrt b } \right)}} + \dfrac{{ - \left( {\sqrt a - \sqrt b } \right)}}{{\left( {\sqrt a - \sqrt b } \right)\left( {\sqrt a + \sqrt b } \right)}}\\ Q = \dfrac{1}{{\sqrt a + \sqrt b }} + \dfrac{{ - 1}}{{\sqrt a + \sqrt b }} = 0 \)
Vậy Q không phụ thuộc vào a,b