a: \(=\dfrac{\sqrt{2}\left(2\sqrt{2}+3\right)+2\sqrt{2}-3}{-1}\)

\(=\dfrac{4+3\sqrt{2}+2\sqrt{2}-3}{-1}=-1-5\sqrt{2}\)

b: \(=\dfrac{1}{\sqrt{10}+\sqrt{6}}-\dfrac{1}{\sqrt{10}-\sqrt{6}}\)

\(=\dfrac{\sqrt{10}-\sqrt{6}-\sqrt{10}-\sqrt{6}}{4}=\dfrac{-2\sqrt{6}}{4}=-\dfrac{\sqrt{6}}{2}\)

c: \(\dfrac{-2}{3\sqrt{8}}+\dfrac{1}{3-2\sqrt{2}}\)

\(=\dfrac{-2\left(3-2\sqrt{2}\right)+6\sqrt{2}}{6\sqrt{2}\left(3-2\sqrt{2}\right)}=\dfrac{-6+4\sqrt{2}+6\sqrt{2}}{6\sqrt{2}\left(3-2\sqrt{2}\right)}\)

\(=\dfrac{10\sqrt{2}-6}{6\sqrt{2}\left(3-2\sqrt{2}\right)}=\dfrac{10-3\sqrt{2}}{6\left(3-2\sqrt{2}\right)}=\dfrac{18+11\sqrt{2}}{6}\)

Ta có: \(C=\dfrac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

\(=\dfrac{\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{4}+\sqrt{6}+\sqrt{8}}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

\(=\dfrac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)\left(1+\sqrt{2}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

\(=1+\sqrt{2}\)

Ta có: \(B=\dfrac{\sqrt{2-\sqrt{3}}+\sqrt{4-\sqrt{15}}+\sqrt{10}}{\sqrt{23-3\sqrt{5}}}\)

\(=\dfrac{\sqrt{4-2\sqrt{3}}+\sqrt{8-2\sqrt{15}}+2\sqrt{5}}{3\sqrt{5}-1}\)

\(=\dfrac{\sqrt{3}-1+\sqrt{5}-\sqrt{3}+2\sqrt{5}}{3\sqrt{5}-1}\)

=1

2:

ĐKXĐ: x>=3

 \(\Leftrightarrow\sqrt{x-3+2\cdot\sqrt{x-3}\cdot\sqrt{3}+3}+\sqrt{x-3-2\cdot\sqrt{x-3}\cdot\sqrt{3}+3}=2\sqrt{3}\)

=>\(\left|\sqrt{x-3}+\sqrt{3}\right|+\left|\sqrt{x-3}-\sqrt{3}\right|=2\sqrt{3}\)

\(\Leftrightarrow\sqrt{x-3}+\sqrt{3}+\left|\sqrt{x-3}-\sqrt{3}\right|=2\sqrt{3}\)

\(\Leftrightarrow\sqrt{x-3}+\left|\sqrt{x-3}-\sqrt{3}\right|=\sqrt{3}\)(1)

TH1: x>=6

(1) trở thành \(\sqrt{x-3}+\sqrt{x-3}-\sqrt{3}=\sqrt{3}\)

=>\(2\sqrt{x-3}=2\sqrt{3}\)

=>x-3=3

=>x=6(nhận)

TH2: 3<=x<6

Phương trình (1) sẽ là;

\(\sqrt{x-3}+\sqrt{3}-\sqrt{x-3}=\sqrt{3}\)

=>\(\sqrt{3}=\sqrt{3}\)(luôn đúng)

1:

\(A^2=8+2\sqrt{10+2\sqrt{5}}+8-2\sqrt{10+2\sqrt{5}}+2\cdot\sqrt{8^2-\left(2\sqrt{10+2\sqrt{5}}\right)^2}\)

\(=16+2\cdot\sqrt{64-4\cdot\left(10+2\sqrt{5}\right)}\)

\(=16+2\cdot\sqrt{24-8\sqrt{5}}\)

\(=16+2\cdot\sqrt{20-2\cdot2\sqrt{5}\cdot2+4}\)

\(=16+2\cdot\sqrt{\left(2\sqrt{5}-2\right)^2}\)

\(=16+2\cdot\left(2\sqrt{5}-2\right)=12+4\sqrt{5}\)

\(=10+2\cdot\sqrt{10}\cdot\sqrt{2}+2\)

\(=\left(\sqrt{10}+\sqrt{2}\right)^2\)

=>\(A=\sqrt{10}+\sqrt{2}\)

1:

\(A=\sqrt{x^2+\dfrac{2x^2}{3}}=\sqrt{\dfrac{5x^2}{3}}=\left|\sqrt{\dfrac{5}{3}}x\right|=-x\sqrt{\dfrac{5}{3}}\)

2: \(=\left(\dfrac{\sqrt{100}+\sqrt{40}}{\sqrt{5}+\sqrt{2}}+\sqrt{6}\right)\cdot\dfrac{2\sqrt{5}-\sqrt{6}}{2}\)

\(=\dfrac{\left(2\sqrt{5}+\sqrt{6}\right)\left(2\sqrt{5}-\sqrt{6}\right)}{2}\)

\(=\dfrac{20-6}{2}=7\)

a/ \(\sqrt{2}+\sqrt{6}\)

b/ Sửa đề:

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

c/ \(1+\sqrt{2}+\sqrt{5}\)

giải rõ ra hộ mình với

\(B=\sqrt{5}\left(\sqrt{20}-\sqrt{8}\right)+2\sqrt{10}\)

\(=\sqrt{100}-\sqrt{40}+2\sqrt{10}\)

\(=\sqrt{10^2}-\sqrt{2^2.10}+2\sqrt{10}\)

\(=10-2\sqrt{10}+2\sqrt{10}\)

\(=10+0\)

\(=10\)

Sửa đề: \(A=\frac{\sqrt{x}}{\sqrt{x}-2}+\frac{x-3\sqrt{x}+8}{x-7\sqrt{x}+10}-\frac{\sqrt{x}-1}{\sqrt{x}-5}\)

a: ĐKXĐ: x>=0; x<>4; x<>25

Ta có: \(A=\frac{\sqrt{x}}{\sqrt{x}-2}+\frac{x-3\sqrt{x}+8}{x-7\sqrt{x}+10}-\frac{\sqrt{x}-1}{\sqrt{x}-5}\)

\(=\frac{\sqrt{x}}{\sqrt{x}-2}+\frac{x-3\sqrt{x}+8}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-5\right)}-\frac{\sqrt{x}-1}{\sqrt{x}-5}\)

\(=\frac{\sqrt{x}\left(\sqrt{x}-5\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-5\right)}+\frac{x-3\sqrt{x}+8}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-5\right)}-\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-5\right)\left(\sqrt{x}-2\right)}\)

\(=\frac{x-5\sqrt{x}+x-3\sqrt{x}+8-x+3\sqrt{x}-2}{\left(\sqrt{x}-5\right)\left(\sqrt{x}-2\right)}=\frac{x-5\sqrt{x}+6}{\left(\sqrt{x}-5\right)\left(\sqrt{x}-2\right)}\)

\(=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-5\right)\left(\sqrt{x}-2\right)}=\frac{\sqrt{x}-3}{\sqrt{x}-5}\)

b: Để A nguyên thì \(\sqrt{x}-3\) ⋮\(\sqrt{x}-5\)

=>\(\sqrt{x}-5+2\vdots\sqrt{x}-5\)

=>\(2\vdots\sqrt{x}-5\)

=>\(\sqrt{x}-5\in\left\lbrace1;-1;2;-2\right\rbrace\)

=>\(\sqrt{x}\in\left\lbrace6;4;7;3\right\rbrace\)

=>x∈{36;16;49;9}