Dạng 1: Căn bậc hai có đáp án

Đáp án D

Đáp án B

Đáp án A

Đáp án C

a) $A=\left(\frac{1}{x+\sqrt{x}}-\frac{1}{\sqrt{x}+1}\right):\frac{\sqrt{x}}{x+2\sqrt{x}+1}$

$=\frac{1-\sqrt{x}}{\sqrt{x}.\left(\sqrt{x}+1\right)}\cdot \frac{{\left(\sqrt{x}+1\right)}^2}{\sqrt{x}}=\frac{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}\right)}{\sqrt{x}.\sqrt{x}}=\frac{1-x}{x}$

b) $A>\frac{1}{2}\iff \frac{1-x}{x}>\frac{1}{2}\iff \frac{1-x}{x}-\frac{1}{2}>0\iff \frac{2-2x-x}{2x}>0$

$\iff \frac{2-3x}{2x}>0$ mà $x>0\iff 2-3x>0\iff x<\frac{2}{3}$

Vậy $0<x<\frac{2}{3}$ thì $A>\frac{1}{2}\cdot$

$P=\sqrt{16}-\sqrt[3]{8}+\frac{\sqrt{12}}{\sqrt{3}}=4-2+\sqrt{4}=4-2+2=4$

Với  $a>0;a\ne 1$

$B=\left[\frac{6}{a-1}+\frac{10-2\sqrt{a}}{(a-1)(\sqrt{a}-1)}\right].\frac{{(\sqrt{a}-1)}^2}{4\sqrt{a}}$$=\frac{4\sqrt{a}+4}{(a-1)(\sqrt{a}-1)}.\frac{{(\sqrt{a}-1)}^2}{4\sqrt{a}}$

$\begin{array}{l}\text{A=2}\sqrt{20}+3\sqrt{45}-4\sqrt{80}=2\sqrt{4.5}+3\sqrt{9.5}-4\sqrt{16.5}\\ =2.2\sqrt{5}+3.3\sqrt{5}-4.4\sqrt{5}\\ =4\sqrt{5}+9\sqrt{5}-16\sqrt{5}=-3\sqrt{5}\end{array}$

$\begin{array}{l}B=\left(2+\frac{1}{\sqrt{x}-1}\right).\frac{x-1}{2\sqrt{x}-1}(x\ge 0;x\ne \frac{1}{4})\\ =\frac{2\sqrt{x}-2+1}{\sqrt{x}-1}.\frac{\left(\sqrt{x}-1\right).\left(\sqrt{x}+1\right)}{2\sqrt{x}-1}=\frac{2\sqrt{x}-1}{2\sqrt{x}-1}.\left(\sqrt{x}+1\right)=\sqrt{x}+1\end{array}$

$\text{A=}\sqrt{45}+\sqrt{20}-2\sqrt{5}=3\sqrt{5}+2\sqrt{5}-2\sqrt{5}=3\sqrt{5}$

$B=\frac{a+2\sqrt{a}}{\sqrt{a}+2}-\frac{a-4}{\sqrt{a}-2}=\frac{\sqrt{a}(\sqrt{a}+2)}{\sqrt{a}+2}-\frac{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}{\sqrt{a}-2}$

$=\sqrt{a}-\sqrt{a}-2=-2$

$A=\frac{1}{2-\sqrt{3}}=\frac{2+\sqrt{3}}{4-3}=2+\sqrt{3}.$

$\frac{\sqrt{a}}{\sqrt{a}+2}+\frac{2(\sqrt{a}-2)}{a-4}=\frac{\sqrt{a}(\sqrt{a}-2)+2(\sqrt{a}-2)}{a-4}=\frac{a-2\sqrt{a}+2\sqrt{a}-4}{a-4}=\frac{a-4}{a-4}=1.$

$A=\left[\frac{x\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{x-1}{\sqrt{x}-1}\right]:\left(\sqrt{x}+\frac{\sqrt{x}}{\sqrt{x}-1}\right)$ $=\left[\frac{\sqrt{x^3}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}-1}\right]:\frac{x-\sqrt{x}+\sqrt{x}}{\sqrt{x}-1}$ $=\left[\frac{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\sqrt{x}-1\right].\frac{\sqrt{x}-1}{x}$$=\left(\frac{x-\sqrt{x}+1}{\sqrt{x}-1}+\frac{x-1}{\sqrt{x}-1}\right).\frac{\sqrt{x}-1}{x}$$=\frac{x-\sqrt{x}+1+x-1}{\sqrt{x}-1}.\frac{\sqrt{x}-1}{x}$$=\frac{x-\sqrt{x}+1+x-1}{\sqrt{x}-1}.\frac{\sqrt{x}-1}{x}$$=\frac{2x-\sqrt{x}}{x}=\frac{2\sqrt{x}-1}{\sqrt{x}}$

b. $A=3\iff \frac{2\sqrt{x}-1}{\sqrt{x}}=3\iff \sqrt{x}-1=3\sqrt{x}\iff \sqrt{x}=-1$

$S=\frac{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}-\frac{a-\sqrt{a}+1}{\sqrt{a}}=\frac{a+\sqrt{a}+1}{\sqrt{a}}-\frac{a-\sqrt{a}+1}{\sqrt{a}}=\frac{2\sqrt{a}}{\sqrt{a}}=2$

$H=\sqrt{81}-\sqrt{16}=9-4=5$

$M=\left(\frac{x+\sqrt{y}+\sqrt{xy}-1}{\sqrt{x}+1}+1\right).\left(\sqrt{x}-\sqrt{y}\right)$$=\frac{x+\sqrt{y}+\sqrt{xy}-1+\sqrt{x}+1}{\sqrt{x}+1}.\left(\sqrt{x}-\sqrt{y}\right)=\frac{x+\sqrt{y}+\sqrt{xy}+\sqrt{x}}{\sqrt{x}+1}.\left(\sqrt{x}-\sqrt{y}\right)$$=\frac{\left(x+\sqrt{xy}\right)+\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}+1}.\left(\sqrt{x}-\sqrt{y}\right)=\frac{\sqrt{x}\left(\sqrt{x}+\sqrt{y}\right)+\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}+1}.\left(\sqrt{x}-\sqrt{y}\right)$$=\frac{\left(\sqrt{x}+1\right).\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}+1}.\left(\sqrt{x}-\sqrt{y}\right)=\left(\sqrt{x}+\sqrt{y}\right).\left(\sqrt{x}-\sqrt{y}\right)=x-y$

$A=2:\left(\frac{1}{\sqrt{\sqrt{x}+1}-1}-\frac{1}{\sqrt{\sqrt{x}+1}+1}\right)$$=2:\frac{\sqrt{\sqrt{x}+1}+1-\sqrt{\sqrt{x}+1}+1}{\left(\sqrt{\sqrt{x}+1}-1\right)\left(\sqrt{\sqrt{x}+1}+1\right)}=2:\frac{2}{\sqrt{x}+1-1}=2.\frac{\sqrt{x}}{2}=\sqrt{x}$

$\text{A=}\sqrt{12}+\sqrt{3}=\sqrt{4.3}+\sqrt{3}=2\sqrt{3}+\sqrt{3}=3\sqrt{3}$

$B={\left(x+2\right)}^2-x^2=\left(x+2-x\right)\left(x+2+x\right)=2(2x+2)=4x+4$

$\begin{array}{l}E=2\sqrt{48}+3\sqrt{75}-2\sqrt{108}=2\sqrt{16.3}+3\sqrt{25.3}-2\sqrt{36.3}\\ =2.4\sqrt{3}+3.5\sqrt{3}-2.6\sqrt{3}=8\sqrt{3}+15\sqrt{3}-12\sqrt{3}=11\sqrt{3}\end{array}$

$\begin{array}{l}P\left(x\right)=\left(\frac{1}{x^2-x}+\frac{1}{x-1}\right):\frac{x+1}{x^2-2x+1}\\ =\left(\frac{1}{x(x-1)}+\frac{x}{x(x-1)}\right):\frac{x+1}{{(x-1)}^2}=\frac{1+x}{x(x-1)}.\frac{{(x-1)}^2}{x+1}=\frac{x-1}{x}\end{array}$

$\left(\sqrt{3}+1\right).\frac{\sqrt{3}-3}{2\sqrt{3}}=\left(\sqrt{3}+1\right).\frac{\sqrt{3}(1-\sqrt{3})}{2\sqrt{3}}=-\frac{\left(\sqrt{3}+1\right).\left(\sqrt{3}-1\right)}{2}=-\frac{3-1}{2}=-1$

$\begin{array}{l}\text{A=}\left(1-\frac{2\sqrt{x}}{3\sqrt{x}+1}-\frac{1-2\sqrt{x}}{1-9x}\right):\left(\frac{6\sqrt{x}+5}{3\sqrt{x}+1}-2\right)\\ =\frac{9x-1-2\sqrt{x}(3\sqrt{x}-1)+1-2\sqrt{x}}{\left(3\sqrt{x}+1\right)\left(3\sqrt{x}-1\right)}:\frac{6\sqrt{x}+5-2(3\sqrt{x}+1)}{3\sqrt{x}+1}\\ =\frac{9x-2\sqrt{x}-6x+2\sqrt{x}}{\left(3\sqrt{x}+1\right)\left(3\sqrt{x}-1\right)}.\frac{3\sqrt{x}+1}{6\sqrt{x}+5-6\sqrt{x}-2}\\ =\frac{3x}{3\sqrt{x}-1}.\frac{1}{3}=\frac{x}{3\sqrt{x}-1}\end{array}$

$\begin{array}{l}a)A=\frac{\sqrt{x}}{\sqrt{x}+3}+\frac{2\sqrt{x}}{\sqrt{x}-3}-\frac{3x+9}{x-9}\\ =\frac{\sqrt{x}}{\sqrt{x}+3}+\frac{2\sqrt{x}}{\sqrt{x}-3}-\frac{3x+9}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}=\frac{\sqrt{x}\left(\sqrt{x}-3\right)+2\sqrt{x}\left(\sqrt{x}+3\right)-3x-9}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}\\ =\frac{x-3\sqrt{x}+2x+6\sqrt{x}-3x-9}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}=\frac{3\sqrt{x}-9}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}=\frac{3\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}=\frac{3}{\sqrt{x}+3}\\ \\ \end{array}$

$B=\sqrt{3}\left(\sqrt{3^2.3}-2\sqrt{2^2.3}+\sqrt{4^2.3}\right)=\sqrt{3}\left(3\sqrt{3}-2.2\sqrt{3}+4\sqrt{3}\right)=\sqrt{3}.3\sqrt{3}=9$

$c)C=\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{\sqrt{a}}{a-\sqrt{a}}\right):\frac{\sqrt{a}+1}{a-1}$$=\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}\right):\frac{\sqrt{a}+1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}$$=\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{1}{\sqrt{a}-1}\right):\frac{1}{\sqrt{a}-1}=\sqrt{a}-1.$

$A=\sqrt{4-2\sqrt{3}}-\frac{1}{2}\sqrt{12}=\sqrt{\left(\sqrt{3}-1\right)}-\frac{1}{2}.2\sqrt{3}=\left|\sqrt{3}-1\right|-\sqrt{3}=\sqrt{3}-1-\sqrt{3}=-1$

$\begin{array}{l}Q\left(x\right)=\frac{5x^2+6x+2018}{x+1}=\frac{5x\left(x+1\right)+x+1+2017}{x+1}=5x+1+\frac{2017}{x+1}\\ \end{array}$

$\left(\sqrt{3}+1\right).\frac{\sqrt{3}-3}{2\sqrt{3}}=\left(\sqrt{3}+1\right).\frac{\sqrt{3}(1-\sqrt{3})}{2\sqrt{3}}=-\frac{\left(\sqrt{3}+1\right).\left(\sqrt{3}-1\right)}{2}=-\frac{3-1}{2}=-1$

$\begin{array}{l}\text{A=}\left(1-\frac{2\sqrt{x}}{3\sqrt{x}+1}-\frac{1-2\sqrt{x}}{1-9x}\right):\left(\frac{6\sqrt{x}+5}{3\sqrt{x}+1}-2\right)\\ =\frac{9x-1-2\sqrt{x}(3\sqrt{x}-1)+1-2\sqrt{x}}{\left(3\sqrt{x}+1\right)\left(3\sqrt{x}-1\right)}:\frac{6\sqrt{x}+5-2(3\sqrt{x}+1)}{3\sqrt{x}+1}\\ =\frac{9x-2\sqrt{x}-6x+2\sqrt{x}}{\left(3\sqrt{x}+1\right)\left(3\sqrt{x}-1\right)}.\frac{3\sqrt{x}+1}{6\sqrt{x}+5-6\sqrt{x}-2}\\ =\frac{3x}{3\sqrt{x}-1}.\frac{1}{3}=\frac{x}{3\sqrt{x}-1}\end{array}$

$\begin{array}{l}a)A=\frac{\sqrt{x}}{\sqrt{x}+3}+\frac{2\sqrt{x}}{\sqrt{x}-3}-\frac{3x+9}{x-9}\\ =\frac{\sqrt{x}}{\sqrt{x}+3}+\frac{2\sqrt{x}}{\sqrt{x}-3}-\frac{3x+9}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}=\frac{\sqrt{x}\left(\sqrt{x}-3\right)+2\sqrt{x}\left(\sqrt{x}+3\right)-3x-9}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}\\ =\frac{x-3\sqrt{x}+2x+6\sqrt{x}-3x-9}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}=\frac{3\sqrt{x}-9}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}=\frac{3\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right).\left(\sqrt{x}+3\right)}=\frac{3}{\sqrt{x}+3}\\ \\ \end{array}$

$B=\sqrt{3}\left(\sqrt{3^2.3}-2\sqrt{2^2.3}+\sqrt{4^2.3}\right)=\sqrt{3}\left(3\sqrt{3}-2.2\sqrt{3}+4\sqrt{3}\right)=\sqrt{3}.3\sqrt{3}=9$

$C=\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{\sqrt{a}}{a-\sqrt{a}}\right):\frac{\sqrt{a}+1}{a-1}$$=\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}\right):\frac{\sqrt{a}+1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}$$=\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{1}{\sqrt{a}-1}\right):\frac{1}{\sqrt{a}-1}=\sqrt{a}-1.$

$A=\sqrt{4-2\sqrt{3}}-\frac{1}{2}\sqrt{12}=\sqrt{\left(\sqrt{3}-1\right)}-\frac{1}{2}.2\sqrt{3}=\left|\sqrt{3}-1\right|-\sqrt{3}=\sqrt{3}-1-\sqrt{3}=-1$