Ta có: \(\frac{1+\sqrt{x}}{1-\sqrt{x}}+\frac{1-\sqrt{x}}{1+\sqrt{x}}\)

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

\(=\frac{x+2\sqrt{x}+1+x-2\sqrt{x}+1}{1-x}=\frac{2x+2}{1-x}\)

\(\dfrac{\sqrt{x}}{\sqrt{x}-1}-\dfrac{2\sqrt{x}}{x-1}-\dfrac{1}{\sqrt{x}+1}\left(đk:x\ne1,x\ge0\right)\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)-2\sqrt{x}-\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\dfrac{x+\sqrt{x}-2\sqrt{x}-\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\)

ĐKXĐ: \(x\ne1,x\ge0\)

\(\dfrac{\sqrt{x}}{\sqrt{x}-1}-\dfrac{2\sqrt{x}}{x-1}-\dfrac{1}{\sqrt{x}+1}=\)\(\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{x-1}-\dfrac{2\sqrt{x}}{x-1}-\dfrac{\sqrt{x}-1}{x-1}=\)\(\dfrac{x+\sqrt{x}-2\sqrt{x}-\sqrt{x}+1}{x-1}=\)\(\dfrac{x-2\sqrt{x}+1}{x-1}=\)\(\dfrac{(\sqrt{x}-1)^2}{x-1}=\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\)

A=1+3+3² +...+3²⁹⁹ +3³⁰⁰

3A=3+3² +...+3²⁹⁹ +3³⁰⁰ +3³⁰¹

3A-A=3³⁰¹ - 1

=> A=\(\frac{3^{^{301}}-1}{2}\)

3√2 - 5√18 + 6√72 - 4√98 = 3√2-5.3√2+6.2.3√2-4.7/3.3√2

                                          = 3√2(1-5+12-28/3)
                                          = 3√2.(-4/3)
                                          = -4√2

\(A=\left(\dfrac{\sqrt{x}-2}{x-1}-\dfrac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right)\cdot\dfrac{\left(x-1\right)^2}{2}\)

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

\(=-\sqrt{x}\cdot\left(\sqrt{x}+1\right)\)

Điều kiện xác định : \(x\ge0;y\ge0;\sqrt{x}-\sqrt{y}\ne-3\)

Ta có : \(\frac{x-y+3\sqrt{x}+3\sqrt{y}}{\sqrt{x}-\sqrt{y}+3}=\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)+3\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}+3}=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}+3\right)}{\sqrt{x}-\sqrt{y}+3}=\sqrt{x}+\sqrt{y}\)

     2\(\sqrt{\dfrac{16}{3}}\)  - 3\(\sqrt{\dfrac{1}{27}}\) - \(\dfrac{3}{2\sqrt{3}}\)

= \(\dfrac{8}{\sqrt{3}}\) - \(\dfrac{3}{3\sqrt{3}}\)  - \(\dfrac{3}{2\sqrt{3}}\)

= \(\dfrac{8}{\sqrt{3}}\) - \(\dfrac{1}{\sqrt{3}}\) - \(\dfrac{3}{2\sqrt{3}}\)

= \(\dfrac{16}{2\sqrt{3}}\) - \(\dfrac{2}{2\sqrt{3}}\) - \(\dfrac{3}{2\sqrt{3}}\)

= \(\dfrac{11}{2\sqrt{3}}\)

= \(\dfrac{11\sqrt{3}}{6}\)

f, 2\(\sqrt{\dfrac{1}{2}}\)- \(\dfrac{2}{\sqrt{2}}\) + \(\dfrac{5}{2\sqrt{2}}\)

= \(\dfrac{2}{\sqrt{2}}\) - \(\dfrac{2}{\sqrt{2}}\) + \(\dfrac{5}{2\sqrt{2}}\)

= \(\dfrac{5}{2\sqrt{2}}\)

= \(\dfrac{5\sqrt{2}}{4}\)

(1 + \(\dfrac{3-\sqrt{3}}{\sqrt{3}-1}\)).(1- \(\dfrac{3+\sqrt{3}}{\sqrt{3}+1}\)) 

= \(\dfrac{\sqrt{3}-1+3-\sqrt{3}}{\sqrt{3}-1}\).\(\dfrac{\sqrt{3}+1-3+\sqrt{3}}{\sqrt{3}+1}\)

= \(\dfrac{2}{\sqrt{3}-1}\).\(\dfrac{-2}{\sqrt{3}+1}\)

= \(\dfrac{-4}{3-1}\)

= \(\dfrac{-4}{2}\)

= -2

5.7.11.13.37-10101/1212120+40404

=3.5.7.11.13.37-10101/1212120.1/10+40404 (vì 1/1212120=1/121212.1/10)

= 3.37.7.11.13.5-101010/121212.1/100+40404

=111.1001.5-5/6.1/100+40404

=151515.5-250/3

=595959-250/3

=1787876/3

Lời giải:
$A=\cos 2x-2\sin 5x\sin x=\cos 2x-2.\frac{-1}{2}[\cos (5x+x)-\cos (5x-x)]$

$=\cos 2x+\cos 6x-\cos 4x$

$=(\cos 2x+\cos 6x)-\cos 4x$

$=2\cos \frac{2x+6x}{2}\cos \frac{6x-2x}{2}-\cos 4x$

$=2\cos 4x\cos 2x-\cos 4x$

$=\cos 4x[2\cos 2x-1]$

Những đáp án A,B,C,D bạn đưa ra không có đáp án nào đúng cả.