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\(\dfrac{\sqrt{x-1}}{x^2}\)
ĐKXĐ: \(\left\{{}\begin{matrix}x-1\ge0\\x^2\ne0\end{matrix}\right.\Leftrightarrow x\ge1\)
\(\sqrt{\dfrac{x}{\left(x-1\right)^2}}\)
ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\x-1\ne0\end{matrix}\right.\) \(\Leftrightarrow x\ge0\)
\(\sqrt{x+5}-\sqrt{2x+1}\)
ĐKXĐ:\(\left\{{}\begin{matrix}x+5\ge0\\2x+1\ge0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-5\\x\ge\dfrac{-1}{2}\end{matrix}\right.\)\(\Leftrightarrow x\ge\dfrac{-1}{2}\)
\(\sqrt{3-x^2}\)
ĐKXĐ: \(3-x^2\ge0\Leftrightarrow x\le\pm\sqrt{3}\)
a, ĐKXĐ: \(2-4x\ge0\)
\(\Rightarrow x\le\dfrac{1}{2}\)
b, ĐKXĐ: \(\left\{{}\begin{matrix}\dfrac{-3}{x-1}>0\\x^2+4\ge0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x-1< 0\\x\in R\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x< 1\\x\in R\end{matrix}\right.\)
(Do ta có: \(x^2+4\ge0\) \(\left(\forall x\in R\right)\))
c, ĐKXĐ: \(4x^2-12x+9>0\) (do biểu thức căn dưới mẫu)
\(\Rightarrow\left(2x-3\right)^2>0\)
\(\Rightarrow x\ne\dfrac{3}{2}\)
\(\text{a) }\dfrac{1}{\sqrt{x-1}}\\ \text{Để biểu thức có nghĩa }\\ thì\Rightarrow\left\{{}\begin{matrix}x-1\ge0\\\sqrt{x-1}\ne0\end{matrix}\right.\\ \Rightarrow x-1>0\\ \Rightarrow x>1\)
\(\text{b) }\dfrac{1}{\sqrt{x-\sqrt{2x-1}}}\\ \text{Để biểu thức có nghĩa }\\ thì\Rightarrow\left\{{}\begin{matrix}x-\sqrt{2x-1}\ge0\\\sqrt{x-\sqrt{2x-1}}\ne0\end{matrix}\right.\\ \Rightarrow x-\sqrt{2x-1}>0\\ \Rightarrow x>\sqrt{2x-1}\\ \Rightarrow x^2>2x-1\\ \Rightarrow x^2-2x+1>0\\ \Rightarrow\left(x-1\right)^2>0\\ \Rightarrow\left|x-1\right|>0\\ \Rightarrow\left[{}\begin{matrix}x-1< 0\\x-1>0\end{matrix}\right.\\ \Rightarrow x-1\ne0\\ \Rightarrow x\ne1\)
\(c\text{) }\sqrt{-\dfrac{1}{x}}\\ \text{Để biểu thức có nghĩa }\\ thì\Rightarrow\left\{{}\begin{matrix}-\dfrac{1}{x}\ge0\\x\ne0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}\dfrac{1}{x}< 0\\x\ne0\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}x< 0\left(\text{Vì }1>0\right)\\x\ne0\end{matrix}\right.\Rightarrow x< 0\)
\(\text{d) }\sqrt{\dfrac{a+1}{a^2}}\\ \text{Để biểu thức có nghĩa }\\ thì\Rightarrow\left\{{}\begin{matrix}\dfrac{a+1}{a^2}\ge0\\a^2\ne0\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}a+1\ge0\left(\text{Vì }a^2>0\right)\\a\ne0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a\ge-1\\a\ne0\end{matrix}\right.\)
Bài 3:
a: \(=\left(4\sqrt{2}-6\sqrt{2}\right)\cdot\dfrac{\sqrt{2}}{2}=-2\sqrt{2}\cdot\dfrac{\sqrt{2}}{2}=-2\)
b: \(=\dfrac{\sqrt{6}\left(\sqrt{3}-\sqrt{2}\right)}{\sqrt{3}-\sqrt{2}}-2\left(\sqrt{6}-1\right)\)
\(=\sqrt{6}-2\sqrt{6}+2=2-\sqrt{6}\)
a) ĐKXĐ: \(\left\{{}\begin{matrix}x\ne1\\x>0\end{matrix}\right.\)
b)
\(D=\left(\dfrac{2x+1}{\sqrt{x^3}-1}-\dfrac{\sqrt{x}}{x+\sqrt{x}+1}\right)\left(\dfrac{1+\sqrt{x^3}}{1+\sqrt{x}}-\sqrt{x}\right)\)
\(=\left(\dfrac{2x+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right)\left(\dfrac{\left(1+\sqrt{x}\right)\left(1-\sqrt{x}+x\right)}{1+\sqrt{x}}-\sqrt{x}\right)\)
\(=\dfrac{2x+1-x+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\left(1-\sqrt{x}+x-\sqrt{x}\right)\)
\(=\dfrac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\left(\sqrt{x}-1\right)^2\)
\(=\sqrt{x}-1\)
c)
Giả sử \(D>\dfrac{-2}{\sqrt{x}}\)
\(\Rightarrow\sqrt{x}-1>-\dfrac{2}{\sqrt{x}}\Leftrightarrow\sqrt{x}-1+\dfrac{2}{\sqrt{x}}>0\)
\(\Leftrightarrow\dfrac{\left(\sqrt{x}-1\right)\sqrt{x}+2}{\sqrt{x}}>0\Leftrightarrow x-\sqrt{x}+2>0\Leftrightarrow\left(x-\sqrt{x}+\dfrac{1}{4}\right)+\dfrac{7}{4}>0\Leftrightarrow\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{7}{4}>0\)(luôn đúng)
b: \(P=\left(\sqrt{x}+\sqrt{x-1}-\left(\sqrt{x-1}+\sqrt{2}\right)\right)\cdot\left(\dfrac{2}{\sqrt{2}-\sqrt{x}}-\dfrac{\sqrt{x}+\sqrt{2}}{\sqrt{x}\left(\sqrt{2}-\sqrt{x}\right)}\right)\)
\(=\left(\sqrt{x}-\sqrt{2}\right)\cdot\dfrac{2\sqrt{x}-\sqrt{x}-\sqrt{2}}{\sqrt{x}\left(\sqrt{2}-\sqrt{x}\right)}\)
\(=-\dfrac{\sqrt{x}-\sqrt{2}}{\sqrt{x}}\)
c: Thay \(x=3-2\sqrt{2}\) vào P, ta được:
\(P=-\dfrac{\sqrt{2}-1-\sqrt{2}}{\sqrt{2}-1}=\dfrac{1}{\sqrt{2}-1}=\sqrt{2}+1\)
1. b) \(\left(x\sqrt{\dfrac{6}{x}}+\sqrt{\dfrac{2x}{3}}+\sqrt{6x}\right):\sqrt{6x}\)
=\(\left(x\sqrt{\dfrac{6x}{x^2}}+\sqrt{\dfrac{6x}{9}}+\sqrt{6x}\right):\sqrt{6x}\)
=\(\left(\sqrt{6x}+\dfrac{1}{3}\sqrt{6x}+\sqrt{6x}\right):\sqrt{6x}\)
=\(\dfrac{7}{3}\sqrt{6x}:\sqrt{6x}=\dfrac{7}{3}\)
2.
P=\(\dfrac{\sqrt{x}-3}{\sqrt{x}-2}-\dfrac{2\sqrt{x}-1}{\sqrt{x}-1}+\dfrac{x-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)(bn có ghi sai đề ko)
a) ĐKXĐ : \(x\ge1,x\ge2,x\ge0\)
b) P=\(\dfrac{\left(\sqrt{x}-3\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}-\dfrac{\left(2\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}+\dfrac{x-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)
=\(\dfrac{x-3\sqrt{x}-\sqrt{x}+3-2x+\sqrt{x}+4\sqrt{x}-2+x-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)
=\(\dfrac{\sqrt{x}-1}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}=\dfrac{1}{\sqrt{x}-2}\)
c) thay x= \(4-2\sqrt{3}\)vào P ta có :
\(\dfrac{1}{\sqrt{4-2\sqrt{3}}-2}=\dfrac{1}{\sqrt{3}-1-2}=\dfrac{1}{\sqrt{3}-3}\)
ĐKXĐ: \(x>0;x\ne1\)
\(\dfrac{2x+\sqrt{x}-1}{1-x}+\dfrac{2x\sqrt{x}+x-\sqrt{x}}{1+x\sqrt{x}}=\dfrac{x-1+x+\sqrt{x}}{1-x}+\dfrac{x\sqrt{x}-\sqrt{x}+x\sqrt{x}+x}{1+x\sqrt{x}}\)
\(=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)+\sqrt{x}\left(\sqrt{x}+1\right)}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}\right)}+\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)+x\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\)
\(=\dfrac{2\sqrt{x}-1}{1-\sqrt{x}}+\dfrac{\left(2\sqrt{x}-1\right)\sqrt{x}}{x-\sqrt{x}+1}=\left(2\sqrt{x}-1\right)\left(\dfrac{1}{1-\sqrt{x}}+\dfrac{\sqrt{x}}{x-\sqrt{x}+1}\right)\)
\(=\dfrac{2\sqrt{x}-1}{\left(1-\sqrt{x}\right)\left(x-\sqrt{x}+1\right)}\)
Vậy \(A=\left(\dfrac{2\sqrt{x}-1}{\sqrt{x}\left(1-\sqrt{x}\right)}\right):\left(\dfrac{2\sqrt{x}-1}{\left(1-\sqrt{x}\right)\left(x-\sqrt{x}+1\right)}\right)\)
\(A=\dfrac{x-\sqrt{x}+1}{\sqrt{x}}\)
b/ Dễ dàng nhận ra \(A>0\)\(A=\dfrac{x-\sqrt{x}+1}{\sqrt{x}}=\sqrt{x}-1+\dfrac{1}{\sqrt{x}}=\sqrt{17-12\sqrt{2}}-1+\dfrac{1}{\sqrt{17-12\sqrt{2}}}\)
\(A=\sqrt{17-12\sqrt{2}}-1+\sqrt{17+12\sqrt{2}}=\sqrt{\left(3-2\sqrt{2}\right)^2}-1+\sqrt{\left(3+2\sqrt{2}\right)^2}\)
\(\Rightarrow A=3-2\sqrt{2}+3+2\sqrt{2}-1=6-1=5\)
c/ Ta có \(A=\sqrt{x}+\dfrac{1}{\sqrt{x}}-1>2\sqrt{\sqrt{x}.\dfrac{1}{\sqrt{x}}}-1=1\) (dấu "=" không xảy ra)
Mà \(A>0\Rightarrow\sqrt{A}>1\Rightarrow\sqrt{A}-1>0\)
Ta có \(A-\sqrt{A}=\sqrt{A}\left(\sqrt{A}-1\right)>0\) do \(\left\{{}\begin{matrix}\sqrt{A}>0\\\sqrt{A}-1>0\end{matrix}\right.\)
\(\Rightarrow A>\sqrt{A}\) \(\forall x\)
a, dk \(1-16x^2\ge0\Leftrightarrow\left(1-4x\right)\left(1+4x\right)\ge0\)
\(\Leftrightarrow-\frac{1}{4}\le x\le\frac{1}{4}\)
b tuong tu
c, \(\sqrt{\left(x-3\right)\left(5-x\right)}\ge0\Leftrightarrow\left(x-3\right)\left(5-x\right)\ge0\Leftrightarrow3\le x\le5\)
d.\(\sqrt{x^2-x+1}>0\)
ma \(x^2-x+1=x^2-2.\frac{1}{2}x+\frac{1}{4}+\frac{3}{4}=\left(x-\frac{1}{2}\right)^2+\frac{3}{4}>0\)
suy ra thoa man vs moi x
ĐKXĐ: \(\left\{{}\begin{matrix}2x-1\ge0\\x+2>0\\3-x\ge0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x\ge\dfrac{1}{2}\\x>-2\\x\le3\end{matrix}\right.\)
\(\Rightarrow\dfrac{1}{2}\le x\le3\)