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1a)
\(D=\frac{a^2+\sqrt{a}}{a-\sqrt{a}+1}-\frac{2a+\sqrt{a}}{\sqrt{a}}+1\left(ĐK:a\ge0\right)\)
\(=\frac{\sqrt{a}\left(a\sqrt{a}+1\right)}{a-\sqrt{a}+1}-\frac{\sqrt{a}\left(2\sqrt{a}+1\right)}{\sqrt{a}}+1\)
\(=\frac{\sqrt{a}\left(\sqrt{a}+1\right)\left(a-\sqrt{a}+1\right)}{a-\sqrt{a}+1}-\left(2\sqrt{a}+1\right)+1\)
\(=a+\sqrt{a}-2\sqrt{a}-1+1=a-\sqrt{a}\)
2:
a: \(E=\dfrac{a-4-5-\sqrt{a}-3}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-2\right)}\)
\(=\dfrac{\left(\sqrt{a}-4\right)\left(\sqrt{a}+3\right)}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-2\right)}=\dfrac{\sqrt{a}-4}{\sqrt{a}-2}\)
b: a^2+3a=0
=>a(a+3)=0
=>a=0(nhận) hoặc a=-3(loại)
Khi a=0 thì \(E=\dfrac{-4}{-2}=2\)
a)
\(A=\dfrac{a^2+\sqrt{a}}{a-\sqrt{a}+1}-\dfrac{2a+\sqrt{a}}{\sqrt{a}}+1\) (ĐKXĐ \(a>0\))
\(\Leftrightarrow A=\dfrac{\sqrt{a}\left(\sqrt{a}+1\right)\left(a-\sqrt{a}+1\right)}{a-\sqrt{a}+1}-2\sqrt{a}\)
\(\Leftrightarrow a+\sqrt{a}-2\sqrt{a}=a-\sqrt{a}\) (Với \(a>0\))
b)
Để A = 2 \(\Rightarrow a-\sqrt{a}=2\)
\(\Leftrightarrow\left(\sqrt{a}-2\right)\left(\sqrt{a}+1\right)=0\)
\(\Leftrightarrow a=4\left(tm\right)\)
Vậy a = 4 thì A = 2 .
c)
\(A=a-\sqrt{a}=\left(\sqrt{a}-\dfrac{1}{2}\right)^2-\dfrac{1}{4}\ge-\dfrac{1}{4}\) Với \(\forall a>0\)
Vậy GTNN của A là \(-\dfrac{1}{4}\) khi a = \(\dfrac{1}{4}\) .
a: \(B=\left(\dfrac{2a+\sqrt{a}-1}{1-a}-\dfrac{2a\sqrt{a}+a-\sqrt{a}}{1-a\sqrt{a}}\right)\)
\(=\dfrac{\left(\sqrt{a}+1\right)\left(2\sqrt{a}-1\right)}{\left(\sqrt{a}+1\right)\left(1-\sqrt{a}\right)}-\dfrac{\sqrt{a}\left(2a+\sqrt{a}-1\right)}{\left(1-\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}\)
\(=\dfrac{2a\sqrt{a}+2a+2\sqrt{a}-a-\sqrt{a}-1-2a\sqrt{a}-a+\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}+a\right)}\)
\(=\dfrac{2\sqrt{a}-1}{\left(1-\sqrt{a}\right)\left(1+a+\sqrt{a}\right)}\)
\(A=1-\dfrac{2\sqrt{a}-1}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\cdot\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\)
\(=1-\dfrac{\sqrt{a}}{a+\sqrt{a}+1}=\dfrac{a+1}{a+\sqrt{a}+1}\)
c: \(A-\dfrac{2}{3}=\dfrac{a+1}{a+\sqrt{a}+1}-\dfrac{2}{3}\)
\(=\dfrac{3a+3-2a-2\sqrt{a}-2}{3\left(a+\sqrt{a}+1\right)}=\dfrac{a-2\sqrt{a}+1}{3\left(a+\sqrt{a}+1\right)}>0\)
=>A>2/3
1/ Rút gọn: \(a)3\sqrt{2a}-\sqrt{18a^3}+4\sqrt{\dfrac{a}{2}}-\dfrac{1}{4}\sqrt{128a}\left(a\ge0\right)=3\sqrt{2a}-3a\sqrt{2a}+2\sqrt{2a}-2\sqrt{2a}=3\sqrt{2a}\left(1-a\right)\)b)\(\dfrac{\sqrt{2}-1}{\sqrt{2}+2}-\dfrac{2}{2+\sqrt{2}}+\dfrac{\sqrt{2}+1}{\sqrt{2}}=\dfrac{\sqrt{2}-1-2}{\sqrt{2}+2}+\dfrac{\sqrt{2}+1}{\sqrt{2}}=\dfrac{\sqrt{2}-3}{\sqrt{2}+2}+\dfrac{\sqrt{2}+1}{\sqrt{2}}=\dfrac{\sqrt{2}-3+2+1+2\sqrt{2}}{\sqrt{2}\left(1+\sqrt{2}\right)}=\dfrac{3\sqrt{2}}{\sqrt{2}\left(1+\sqrt{2}\right)}=\dfrac{3}{1+\sqrt{2}}\)c)\(\dfrac{2+\sqrt{5}}{\sqrt{2}+\sqrt{3+\sqrt{5}}}+\dfrac{2-\sqrt{5}}{\sqrt{2}-\sqrt{3-\sqrt{5}}}=\dfrac{\sqrt{2}\left(2+\sqrt{5}\right)}{\left(\sqrt{2}+\sqrt{3+\sqrt{5}}\right)\sqrt{2}}+\dfrac{\sqrt{2}\left(2-\sqrt{5}\right)}{\sqrt{2}\left(\sqrt{2}-\sqrt{3-\sqrt{5}}\right)}=\dfrac{2\sqrt{2}+\sqrt{10}}{2+\sqrt{6+2\sqrt{5}}}+\dfrac{2\sqrt{2}-\sqrt{10}}{2-\sqrt{6-2\sqrt{5}}}=\dfrac{2\sqrt{2}+\sqrt{10}}{2+\sqrt{\left(\sqrt{5}+1\right)^2}}+\dfrac{2\sqrt{2}-\sqrt{10}}{2-\sqrt{\left(\sqrt{5}-1\right)^2}}=\dfrac{\sqrt{2}\left(2+\sqrt{5}\right)}{2+\sqrt{5}+1}+\dfrac{\sqrt{2}\left(2-\sqrt{5}\right)}{2-\sqrt{5}+1}=\dfrac{\sqrt{2}\left(2+\sqrt{5}\right)}{3+\sqrt{5}}+\dfrac{\sqrt{2}\left(2-\sqrt{5}\right)}{3-\sqrt{5}}=\dfrac{\sqrt{2}\left(2+\sqrt{5}\right)\left(3-\sqrt{5}\right)+\sqrt{2}\left(2-\sqrt{5}\right)\left(3+\sqrt{5}\right)}{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}=\dfrac{\sqrt{2}\left(6-2\sqrt{5}+3\sqrt{5}-5+6+2\sqrt{5}-3\sqrt{5}-5\right)}{9-5}=\dfrac{2\sqrt{2}}{4}=\dfrac{1}{\sqrt{2}}\)
Làm nốt nè :3
\(2.a.P=\left(\dfrac{1}{x-\sqrt{x}}+\dfrac{1}{\sqrt{x}-1}\right):\dfrac{\sqrt{x}}{x-2\sqrt{x}+1}=\dfrac{\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}}=\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{x}=\dfrac{x-1}{x}\left(x>0;x\ne1\right)\)\(b.P>\dfrac{1}{2}\Leftrightarrow\dfrac{x-1}{x}-\dfrac{1}{2}>0\)
\(\Leftrightarrow\dfrac{x-2}{2x}>0\)
\(\Leftrightarrow x-2>0\left(do:x>0\right)\)
\(\Leftrightarrow x>2\)
\(3.a.A=\left(\dfrac{\sqrt{a}}{\sqrt{a}-1}-\dfrac{\sqrt{a}}{a-\sqrt{a}}\right):\dfrac{\sqrt{a}+1}{a-1}=\dfrac{\sqrt{a}-1}{\sqrt{a}-1}.\dfrac{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\sqrt{a}+1}=\sqrt{a}-1\left(a>0;a\ne1\right)\)
\(b.Để:A< 0\Leftrightarrow\sqrt{a}-1< 0\Leftrightarrow a< 1\)
Kết hợp với DKXĐ : \(0< a< 1\)
a) \(\dfrac{a\sqrt{b}+b\sqrt{a}}{\sqrt{ab}}:\dfrac{1}{\sqrt{a}-\sqrt{b}}=\dfrac{\sqrt{ab}\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{ab}}:\dfrac{1}{\sqrt{a}-\sqrt{b}}\)
\(=\left(\sqrt{a}+\sqrt{b}\right).\left(\sqrt{a}-\sqrt{b}\right)=a-b\)
b) đề sai rồi nha
c) \(\dfrac{a\sqrt{a}-8+2a-4\sqrt{a}}{a-4}=\dfrac{a\sqrt{a}-4\sqrt{a}+2a-8}{a-4}\)
\(=\dfrac{\sqrt{a}\left(a-4\right)+2\left(a-4\right)}{a-4}=\dfrac{\left(\sqrt{a}+2\right)\left(a-4\right)}{a-4}=\sqrt{a}+2\)
\(A=1+\left(\dfrac{2a+\sqrt{a}-1}{1-a}-\dfrac{2a\sqrt{a}-\sqrt{a}+a}{1-a\sqrt{a}}\right).\dfrac{a-\sqrt{a}}{2\sqrt{a}-1}\\ =1+\left(\dfrac{2a+2\sqrt{a}-\sqrt{a}-1}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}-\dfrac{2a\sqrt{a}-\sqrt{a}+a}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}+a\right)}\right).\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\)
\(=1+\dfrac{2\sqrt{a}-1+2a+2a\sqrt{a}-a-2a\sqrt{a}+\sqrt{a}-a}{-\left(\sqrt{a}-1\right)\left(1+\sqrt{a}+a\right)}\)
\(=1+\dfrac{2\sqrt{a}-1+0}{1+\sqrt{a}+a}.\dfrac{\sqrt{a}\left(-1\right)}{2\sqrt{a}-1}\\ =1+\dfrac{1}{1+\sqrt{a}+a}.\sqrt{a}.\left(-1\right)\)
\(=1-\dfrac{\sqrt{a}}{1+\sqrt{a}+a}\\ =\dfrac{1+\sqrt{a}+a-\sqrt{a}}{1+\sqrt{a}+a}\\ =\dfrac{1+a}{1+\sqrt{a}+a}\)
a: \(P=a+\sqrt{a}-2\sqrt{a}-1+1=a-\sqrt{a}\)
b: Để A=2 thì a-căn a-2=0
=>a=4
c: \(P=a-\sqrt{a}+\dfrac{1}{4}-\dfrac{1}{4}=\left(\sqrt{a}-\dfrac{1}{2}\right)^2-\dfrac{1}{4}>=-\dfrac{1}{4}\)
Dấu = xảy ra khi a=1/4
a) ĐK: `a >0`
`P=(a^2+\sqrta)/(a-\sqrta+1)-(2a+\sqrta)/(\sqrta)+1`
`=(\sqrta(\sqrt(a^3)+1^3))/(a-\sqrta+1)-(\sqrta(2\sqrta+1))/(\sqrta)+1`
`=(\sqrta(\sqrta+1)(a-\sqrta+1))/(a-\sqrta+1)-(2\sqrta+1)+1`
`=a+\sqrta-2\sqrta-1+1`
`=a-\sqrta`
b) `P=a-\sqrta`
`=(\sqrta)^2-2.\sqrta .1/2 + (1/2)^2 -1/4`
`=(\sqrta-1/2)^2 -1/4 ≥ -1/4`
`=> P_(min) =-1/4 <=> a=1/4`