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\(P=\left(\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{1}{x-\sqrt{x}}\right):\left(\frac{1}{\sqrt{x}+1}+\frac{2}{x-1}\right)\)với x >= 0 ; x khác 1
\(=\left(\frac{x-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right):\left(\frac{\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(x-1\right)}\right)\)
\(=\frac{\sqrt{x}+1}{\sqrt{x}}:\frac{1}{x-1}=\frac{\left(\sqrt{x}+1\right)\left(x-1\right)}{\sqrt{x}}\)
số xấu vậy bạn, bạn kiểm tra lại đề nhé
\(P=\left(\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{1}{x-\sqrt{x}}\right):\left(\frac{1}{1+\sqrt{x}}+\frac{2}{x-1}\right)\)
\(P=\frac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\frac{\sqrt{x}-1+2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(P=\frac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\sqrt{x}+1}\)
\(P=\frac{\sqrt{x}-1}{\sqrt{x}}\)
bài 1
P=\(\left(\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}}{x+\sqrt{x}+1}-\frac{1}{\sqrt{x}-1}\right)\)
=\(\left(\frac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{...}-\frac{\left(x+\sqrt{x}+1\right)}{...}\right):\frac{\sqrt{x}-1}{2}\)
=\(\left(\frac{x+2+x-\sqrt{x}-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right):\frac{\sqrt{x}-1}{2}\)
=\(\left(\frac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right):\frac{\sqrt{x}-1}{2}\)
=\(\left(\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right):\frac{\sqrt{x}-1}{2}\)
=\(\frac{\sqrt{x}-1}{x+\sqrt{x}+1}.\frac{2}{\sqrt{x}-1}\)
=\(\frac{2}{x+\sqrt{x}+1}\)
P>0 dựa vào dkxd
Bài 1:
a) P= \(\left(\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}}{x+\sqrt{x}+1}+\frac{1}{1-\sqrt{x}}\right):\frac{\sqrt{x}-1}{2}\) (x ≥ 0; x ≠ 4)
=\(\left(\frac{x+2}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}+\frac{\left(\sqrt{x}-1\right)\cdot\sqrt{x}}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}-\frac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}\right)\cdot\frac{2}{\sqrt{x}-1}\)
= \(\left(\frac{x+2+x-\sqrt{x}-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}\right)\cdot\frac{2}{\sqrt{x}-1}\)
=\(\left(\frac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}\right)\cdot\frac{2}{\sqrt{x}-1}\)
=\(\frac{\left(\sqrt{x}-1\right)^2\cdot2}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)\cdot\left(\sqrt{x}-1\right)}\)
=\(\frac{2}{x+\sqrt{x}+1}\)
b) Ta có: x ≥ 0 ⇒ \(\sqrt{x}\) ≥ 0
⇒ \(x+\sqrt{x}+1\) ≥ 1 > 0
mà 2 > 0 ⇒ \(\frac{2}{x+\sqrt{x}+1}\) > 0 ⇒ P > 0
Bài 2:
a) P= \(\left(\frac{2\sqrt{x}+x}{x\sqrt{x}-1}-\frac{1}{\sqrt{x}-1}\right):\left(1-\frac{\sqrt{x}+2}{x+\sqrt{x}+1}\right)\) (x ≥ 0; x ≠ 1)
=\(\left(\frac{2\sqrt{x}+x}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}-\frac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}\right):\left(\frac{x+\sqrt{x}+1}{x+\sqrt{x}+1}-\frac{\sqrt{x}+2}{x+\sqrt{x}+1}\right)\)
=\(\left(\frac{2\sqrt{x}+x-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}\right):\left(\frac{x+\sqrt{x}+1-\sqrt{x}-2}{x+\sqrt{x}+1}\right)\)
=\(\left(\frac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}\right):\left(\frac{x-1}{x+\sqrt{x}+1}\right)\)
=\(\frac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)\cdot\left(x+\sqrt{x}+1\right)}\cdot\frac{x+\sqrt{x}+1}{x-1}\)
=\(\frac{1}{x-1}\)
b) Ta có: \(\sqrt{P}=\sqrt{\frac{1}{x-1}}\)
= \(\frac{1}{\sqrt{x-1}}\)
x = \(5+2\sqrt{3}\) (TM)
Thay x vào \(\sqrt{P}\) ta có:
\(\sqrt{P}=\frac{1}{\sqrt{5+2\sqrt{3}-1}}\)
=\(\frac{1}{\sqrt{4+2\sqrt{3}}}\)
=\(\frac{1}{\sqrt{3+2\sqrt{x}+1}}\)
=\(\frac{1}{\sqrt{\left(\sqrt{3}+1\right)^2}}\)
=\(\frac{1}{\left|\sqrt{3}+1\right|}\)
=\(\frac{1}{\sqrt{3}+1}\)
= \(\frac{\sqrt{3}-1}{\left(\sqrt{3}+1\right)\cdot\left(\sqrt{3}-1\right)}\)
=\(\frac{\sqrt{3}-1}{2}\)
Vậy \(\sqrt{P}=\frac{\sqrt{3}-1}{2}\) khi x = \(5+2\sqrt{3}\)
a.
\(B=\left(\frac{x+3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}+\frac{\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\right):\frac{\sqrt{x}}{\sqrt{x}-3}\\ =\left(\frac{x+3+\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\right):\frac{\sqrt{x}}{\sqrt{x}-3}\\ =\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\cdot\frac{\sqrt{x}-3}{\sqrt{x}}\\ =\frac{\sqrt{x}+1}{\sqrt{x}+3}\)
b. Ta có :
\(x=\sqrt{27+10\sqrt{2}}-\sqrt{18+8\sqrt{2}}\\ =\sqrt{25+2\cdot5\cdot\sqrt{2}+2}-\sqrt{16+2\cdot4\cdot\sqrt{2}+2}\\ =\sqrt{\left(5+\sqrt{2}\right)^2}-\sqrt{\left(4+\sqrt{2}\right)^2}\\ =5+\sqrt{2}-4-\sqrt{2}=1\)
\(B=\frac{\sqrt{x}+1}{\sqrt{x}+3}=\frac{1+1}{1+3}=\frac{2}{4}=\frac{1}{2}\)
c. Giả sử B>\(\frac{1}{3}\), ta có
\(B=\frac{\sqrt{x}+1}{\sqrt{x}+3}>\frac{1}{3}\\ \Leftrightarrow\frac{\sqrt{x}+1}{\sqrt{x}+3}-\frac{1}{3}>0\\ \Leftrightarrow\\\frac{3\left(\sqrt{x}+1\right)-\left(\sqrt{x}+3\right)}{3\left(\sqrt{x}+3\right)}>0\\ \Leftrightarrow\frac{2\sqrt{x}}{3\left(\sqrt{x}+3\right)}>0\left(luondungvoix>0\right)\)
Vậy.........
\(\frac{1}{2\sqrt{x}-2}-\frac{1}{2\sqrt{x}+2}+\frac{\sqrt{x}}{1-x}\)
ĐKXĐ : \(\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\)
\(=\frac{1}{2\left(\sqrt{x}-1\right)}-\frac{1}{2\left(\sqrt{x}+1\right)}-\frac{\sqrt{x}}{x-1}\)
\(=\frac{\sqrt{x}+1}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\sqrt{x}-1}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{2\sqrt{x}}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{\sqrt{x}+1-\sqrt{x}+1-2\sqrt{x}}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{-2\sqrt{x}+2}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{-2\left(\sqrt{x}-1\right)}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{-1}{\sqrt{x}+1}\)