a: Ta có: \(\sqrt{\left(5-\sqrt{19}\right)^2}-\sqrt{\left(4-\sqrt{19}\right)^2}\)

\(=5-\sqrt{19}-\sqrt{19}+4\)

\(=9-2\sqrt{19}\)

b: Ta có: \(\sqrt{\left(3-2\sqrt{2}\right)^2}-\sqrt{\left(2\sqrt{2}-3\right)^2}\)

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

=0

c.

Căn bậc 2 không xác định do $2-\sqrt{5}< 0$

d.

\(=\sqrt{(3+\sqrt{3})^2}(3+\sqrt{3})=|3+\sqrt{3}|(3+\sqrt{3})=(3+\sqrt{3})^2=12+6\sqrt{3}\)

e.

\(=(2-\sqrt{5})\sqrt{(2+\sqrt{5})^2}=(2-\sqrt{5})|2+\sqrt{5}|=(2-\sqrt{5})(2+\sqrt{5})=4-5=-1\)

a,\(\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2-\sqrt{3}}\)

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

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

\(=\left(\sqrt{3}+1\right)\sqrt{\left(\sqrt{3}-1\right)^2}\)

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

\(=3-1\)

\(=2\)

b, \(\left(\sqrt{4+\sqrt{15}}-\sqrt{16-3\sqrt{15}}\right)\left(\sqrt{3}+\sqrt{5}\right)\)

\(=\frac{\sqrt{8+2\sqrt{15}}-\sqrt{32-6\sqrt{15}}}{\sqrt{2}}.\left(\sqrt{3}+\sqrt{5}\right)\)

\(=\frac{\sqrt{3+2\sqrt{3}.\sqrt{5}+5}-\sqrt{27-2.3\sqrt{3}.\sqrt{5}+5}}{\sqrt{2}}\left(\sqrt{3}+\sqrt{5}\right)\)

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

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

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

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

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

\(=2\sqrt{2}\)

\(A=\dfrac{\sqrt{2+\sqrt{4-x^2}}\left(\sqrt{\left(2+x\right)^3}-\sqrt{\left(2-x\right)^3}\right)}{4+\sqrt{4-x^2}}\)

\(\Rightarrow A=\sqrt{\left(2+x\right)^{^{ }3}}-\sqrt{\left(2-x\right)^3}=\left(\sqrt{2+x}-\sqrt{2-x}\right)\left(4+\sqrt{4-x^2}\right)\)

\(\Rightarrow A=\dfrac{\sqrt{4+2\sqrt{4-x^2}}\left(\sqrt{2+x}-\sqrt{2-x}\right)\left(4+\sqrt{4-x^2}\right)}{\sqrt{2}\left(4+\sqrt{4-x^2}\right)}\)

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

\(2\sqrt{2}\)

\(A=\left(\dfrac{6x+4}{3\sqrt{3x^3}-8}-\dfrac{\sqrt{3x}}{3x+2\sqrt{3x}+4}\right).\left(\dfrac{1+3\sqrt{3x^3}}{1+\sqrt{3x}}-\sqrt{3x}\right)\)

Điều kiện tự làm nha:

Đặt \(\sqrt{3x}=a\) thì ta có:

\(A=\left(\dfrac{2a^2+4}{a^3-8}-\dfrac{a}{a^2+2a+4}\right).\left(\dfrac{1+a^3}{1+a}-a\right)\)

\(=\left(\dfrac{2a^2+4}{\left(a-2\right)\left(a^2+2a+4\right)}-\dfrac{a}{a^2+2a+4}\right).\left(\dfrac{\left(1+a\right)\left(1-a+a^2\right)}{1+a}-a\right)\)

\(=\dfrac{a^2+2a+4}{\left(a-2\right)\left(a^2+2a+4\right)}.\left(1-2a+a^2\right)\)

\(=\dfrac{\left(a-1\right)^2}{a-2}=\dfrac{\left(\sqrt{3x}-1\right)^2}{\sqrt{3x}-2}\)

1/ a/ \(\sqrt{\left(6+2\sqrt{5}\right)^3}-\sqrt{\left(6-2\sqrt{5}\right)^3}\)

\(=\sqrt{\left(\sqrt{5}+1\right)^6}-\sqrt{\left(\sqrt{5}-1\right)^6}\)

\(=\left(\sqrt{5}+1\right)^3-\left(\sqrt{5}-1\right)^3\)

\(=32\)

b/ \(\sqrt{\left(3-2\sqrt{2}\right)\left(4-2\sqrt{3}\right)}\)

\(=\sqrt{\left(\sqrt{2}-1\right)^2\left(\sqrt{3}-1\right)^2}\)

\(=\left(\sqrt{2}-1\right)\left(\sqrt{3}-1\right)\)

\(=\sqrt{6}-\sqrt{2}-\sqrt{3}+1\)

Câu 3/ \(A=\sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt{2+\sqrt{2}}}}}\)

\(< \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt{2+\sqrt{4}}}}}=2\)

Ta lại có:

\(A=\sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt{2+\sqrt{2}}}}}>\sqrt{2}>1\)

\(\Rightarrow1< A< 2\)

Vậy \(A\notin N\)

Câu 2/ Ta có:

\(x=\sqrt{2+\sqrt{2+\sqrt{3}}}-\sqrt{6-3\sqrt{2+\sqrt{3}}}\)

\(\Leftrightarrow x^2=8-2\sqrt{2+\sqrt{3}}-2\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{6-3\sqrt{2+\sqrt{3}}}\)

\(\Leftrightarrow x^2=8-2\sqrt{2+\sqrt{3}}-2\sqrt{3.\left(2+\sqrt{3}\right)}\)

\(\Leftrightarrow x^2-4=4-2\sqrt{2+\sqrt{3}}-2\sqrt{3.\left(2+\sqrt{3}\right)}\)

\(\Leftrightarrow\frac{\left(8-x^2\right)}{2}=\sqrt{2+\sqrt{3}}+\sqrt{3.\left(2+\sqrt{3}\right)}\)

\(\Leftrightarrow\frac{\left(8-x^2\right)^2}{4}=8-2\sqrt{3}+2.\sqrt{2+\sqrt{3}}.\sqrt{3.\left(2-\sqrt{3}\right)}=8-2\sqrt{3}+2\sqrt{3}=8\)

\(\Leftrightarrow\left(x^2-8\right)^2=32\)

Ta có:

\(x^4-16x^2+32=\left(x^4-16x^2+64\right)-32\)

\(=\left(x^2-8\right)^2-32=32-32=0\)

Vậy \(x=\sqrt{2+\sqrt{2+\sqrt{3}}}-\sqrt{6-3\sqrt{2+\sqrt{3}}}\) là nghiệm của phương trình đã cho.

\(A=\left(\frac{3}{\sqrt{x}-1}-\frac{\sqrt{x}-3}{x-1}\right):\left(\frac{x+2}{x+\sqrt{x}-2}-\frac{\sqrt{x}}{\sqrt{x}+2}\right)\left(ĐK:x\ge0;\ne1\right)\)

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

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

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

\(=\frac{2\sqrt{x}+6}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\cdot\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}{\sqrt{x}+2}\)

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