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\(\frac{\sqrt{a}-a}{a\sqrt{a}-a+\sqrt{a}}:\frac{1}{a^2+\sqrt{a}}\)
\(=\frac{\sqrt{a}\left(1-\sqrt{a}\right)}{\sqrt{a}\left(a-\sqrt{a}+1\right)}.\frac{\sqrt{a}\left(a\sqrt{a}+1\right)}{1}\)
\(=\frac{1-\sqrt{a}}{a-\sqrt{a}+1}.\frac{\sqrt{a}\left(\sqrt{a}+1\right)\left(a-\sqrt{a}+1\right)}{1}\)
\(=\sqrt{a}\left(1-\sqrt{a}\right)\left(\sqrt{a}+1\right)\)
\(=\sqrt{a}\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)\)
\(=\sqrt{a}\left(1-a\right)\)
\(=\sqrt{a}-a\sqrt{a}\)
Với 1 ≤ x < 2
A = (x + 3)/2
Với x ≥ 2
A = (x + 3)/[2√(x - 1)]
b/ Xét 1 ≤ x < 2
A ≥ (3 + 1)/2 = 2
Xét x ≥ 2
A = 2 + [√(x - 1) - 2]²/[2√(x - 2)] ≥ 2
Kết hợp 2 TH thì min là 2 khi x = 1 hoặc x = 5
a) \(\sqrt{4\left(a-3\right)^2}=\sqrt{2^2\left(a-3\right)^2}=2\sqrt{\left(a-3\right)^2}=2.\left|a-3\right|=2\left(a-3\right)=2a-6\) (Vì \(a\ge3\) )
b) \(\sqrt{9\left(b-2\right)^2}=\sqrt{3^2\left(b-2\right)^2}=3\sqrt{\left(b-2\right)^2}=3\left|b-2\right|=3\left(2-b\right)\)
\(=6-3b\) (vì b < 2 )
b) \(\sqrt{27.48\left(1-a\right)^2}=\sqrt{27.3.16.\left(1-a\right)^2}=\sqrt{81.16.\left(1-a\right)^2}\)
\(=\sqrt{9^2.4^2.\left(1-a\right)^2}=9.4\sqrt{\left(1-a\right)^2}=36.\left|1-a\right|=36\left(1-a\right)=36-36a\) (vì a > 1)
Mong mọi người giúp đỡ mình , mình đang cần gấp , cảm ơn mọi người
Ta có HĐT : \(\hept{\begin{cases}a\sqrt{a}+b\sqrt{b}=\left(\sqrt{a}+\sqrt{b}\right)\left(a-\sqrt{ab}+b\right)\\a\sqrt{a}-b\sqrt{b}=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)\end{cases}\left(a,b\ge0\right)}\)
\(P=\left(\frac{2a+1}{a\sqrt{a}-1}-\frac{\sqrt{a}}{a+\sqrt{a}+1}\right)\times\left(\frac{1+\sqrt{a^3}}{1+\sqrt{a}}-\sqrt{a}\right)\)
ĐKXĐ : \(\hept{\begin{cases}a\ge0\\a\ne1\end{cases}}\)
\(=\left(\frac{2a+1}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}-\frac{\sqrt{a}}{a+\sqrt{a}+1}\right)\times\left(\frac{1+a\sqrt{a}}{1+\sqrt{a}}-\sqrt{a}\right)\)
\(=\left(\frac{2a+1}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}-\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\right)\times\left(\frac{\left(1+\sqrt{a}\right)\left(1-\sqrt{a}+a\right)}{1+\sqrt{a}}-\sqrt{a}\right)\)
\(=\left(\frac{2a+1-a+\sqrt{a}}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\right)\times\left(1-\sqrt{a}+a-\sqrt{a}\right)\)
\(=\frac{a+\sqrt{a}+1}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\times\left(a-2\sqrt{a}+1\right)\)
\(=\frac{1}{\sqrt{a}-1}\times\left(\sqrt{a}-1\right)^2\)
\(=\frac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}-1}=\sqrt{a}-1\)
b) \(P\times\sqrt{1-a}\)
\(=\left(\sqrt{a}-1\right)\times\sqrt{1-a}\)
ĐKXĐ: \(0\le x< 1\)
Với \(0\le x< 1\)
Ta có :\(\hept{\begin{cases}\sqrt{a}\le\sqrt{1}=1\Rightarrow\sqrt{a}-1\le0\\\sqrt{1-a}\ge0\end{cases}}\)
\(\Rightarrow\left(\sqrt{a}-1\right)\left(\sqrt{1-a}\right)\le0\)
a)
\(A=\frac{\sqrt{a}+3}{\sqrt{a}-2}-\frac{\sqrt{a}-1}{\sqrt{a}+2}+\frac{4\sqrt{a}-4}{4-a}\)
\(=\frac{\left(\sqrt{a}+3\right)\left(\sqrt{a}+2\right)-\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}+\frac{4\sqrt{a}-4}{4-\sqrt{a}}\)
\(=\frac{a+2\sqrt{a}+3\sqrt{a}+6-a-2\sqrt{a}-\sqrt{a}+2}{a-4}+\frac{4\sqrt{a}-4}{4-a}\)
\(=\frac{a-a+\left(2+3-2-1\right)\sqrt{a}+6+2}{a-4}+\frac{-4\sqrt{a}+4}{a-4}\)
\(=\frac{2\sqrt{a}+8}{a-4}+\frac{-4\sqrt{a}+4}{a-4}\)
\(=\frac{2\sqrt{a}+8-4\sqrt{a}+4}{\left(a-4\right)^2}\)
\(=\frac{-2\sqrt{a}+12}{\left(a-4\right)^2}\)
b) thấy A = 9 vào biểu thức , ta có :
\(9=\frac{-2\sqrt{a}+12}{\left(a-4\right)^2}\)
\(< =>\frac{9\left(a-4\right)^2}{\left(a-4\right)^2}=\frac{-2\sqrt{a}+12}{\left(a-4\right)^2}\)
\(< =>9\left(a-4\right)^2=-2\sqrt{a}+12\)
\(< =>9.\left(a^2-2a.4+4^2\right)=-2\sqrt{a}+12\)
\(< =>9a^2-72a+144=-2\sqrt{a}+12\)
\(< =>9a^2-72a+2\sqrt{a}=12-144\)
\(< =>\sqrt{a}\left(9\sqrt{a}^3-72\sqrt{a}+2\right)=-132\)
\(\)
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Điều kiện: x \(\ne\) 1; 1/4 ; x \(\ge\) 0
\(A=1+\left(\frac{\left(2a+\sqrt{a}-1\right)}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}-\frac{\left(2a+\sqrt{a}-1\right).\sqrt{a}}{\left(1-\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}\right)\left(\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\right)\)
\(A=1+\left(\frac{\left(2a+\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)-\left(2a+\sqrt{a}-1\right)\left(1+\sqrt{a}\right).\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}\right)\left(\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\right)\)
\(A=1+\left(\frac{\left(2a+\sqrt{a}-1\right)\left(a+\sqrt{a}+1-a-\sqrt{a}\right)}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}\right)\left(\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\right)\)
\(A=1+\left(\frac{\left(2a+\sqrt{a}-1\right)}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}\right)\left(\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\right)\)
\(A=1+\left(\frac{\left(2\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}\right)\left(\frac{-\sqrt{a}\left(1-\sqrt{a}\right)}{2\sqrt{a}-1}\right)=1+\frac{-\sqrt{a}}{a+\sqrt{a}+1}=\frac{a+1}{a+\sqrt{a}+1}\)
Các bài tập dạng này hoàn toàn làm tương tự!!!
\(P=\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{4-6\sqrt{a}}{1-a}-\frac{-3}{\sqrt{a}+1}\)
ĐK : \(\hept{\begin{cases}a\ge0\\a\ne1\end{cases}}\)
a) \(P=\frac{\sqrt{a}}{\sqrt{a}-1}+\frac{4-6\sqrt{a}}{a-1}+\frac{3}{\sqrt{a}+1}\)
\(=\frac{\sqrt{a}}{\sqrt{a}-1}+\frac{4-6\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+\frac{3}{\sqrt{a}+1}\)
\(=\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+\frac{4-6\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+\frac{3\left(\sqrt{a}-1\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\)
\(=\frac{a+\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+\frac{4-6\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+\frac{3\sqrt{a}-3}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\)
\(=\frac{a+\sqrt{a}+4-6\sqrt{a}+3\sqrt{a}-3}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\)
\(=\frac{a-2\sqrt{a}+1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}=\frac{\left(\sqrt{a}-1\right)^2}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}=\frac{\sqrt{a}-1}{\sqrt{a}+1}\)
Với \(a=4-2\sqrt{3}\)( tmđk \(\hept{\begin{cases}a\ge0\\a\ne1\end{cases}}\))
\(P=\frac{\sqrt{4-2\sqrt{3}}-1}{\sqrt{4-2\sqrt{3}}+1}\)
\(=\frac{\sqrt{3-2\sqrt{3}+1}-1}{\sqrt{3-2\sqrt{3}+1}+1}\)
\(=\frac{\sqrt{\left(\sqrt{3}\right)^2-2\sqrt{3}+1^2}-1}{\sqrt{\left(\sqrt{3}\right)^2-2\sqrt{3}+1^2}+1}\)
\(=\frac{\sqrt{\left(\sqrt{3}-1\right)^2}-1}{\sqrt{\left(\sqrt{3}-1\right)^2}+1}\)
\(=\frac{\left|\sqrt{3}-1\right|-1}{\left|\sqrt{3}-1\right|+1}\)
\(=\frac{\sqrt{3}-1-1}{\sqrt{3}-1+1}=\frac{\sqrt{3}-2}{\sqrt{3}}\)
b) \(P=\frac{\sqrt{a}-1}{\sqrt{a}+1}=\frac{\sqrt{a}+1-2}{\sqrt{a}+1}=1-\frac{2}{\sqrt{a}+1}\)( ĐK \(\hept{\begin{cases}a\ge0\\a\ne1\end{cases}}\))
Để P đạt giá trị nguyên => \(\frac{2}{\sqrt{a}+1}\)nguyên
=> \(2⋮\sqrt{a}+1\)
=> \(\sqrt{a}+1\inƯ\left(2\right)=\left\{\pm1;\pm2\right\}\)
=> \(\sqrt{a}\in\left\{0;1\right\}\)< đã loại hai trường hợp âm >
=> \(a\in\left\{0\right\}\)< loại trường hợp a = 1 >
Vậy với a = 0 thì P có giá trị nguyên
Chọn đáp án C.