\(\sqrt{5\sqrt{5\sqrt{5...\sqrt{5\sqrt{5}}}}}=x\Rightarrow x^2=5x\Rightarrow x=5\)(n số 5)

Vậy \(\sqrt{5\sqrt{5\sqrt{5...\sqrt{5\sqrt{5}}}}}=5\) khi \(n\rightarrow\infty\)

\(\sqrt{6+\sqrt{6+\sqrt{6+...+\sqrt{6+\sqrt{6}}}}}=x\\ \Leftrightarrow x^2=6+\sqrt{6+\sqrt{6+\sqrt{6+...+\sqrt{6+\sqrt{6}}}}}\\ \Leftrightarrow x^2=6+x\Rightarrow x=3\)(n số 6)

\(\sqrt{6+\sqrt{6+\sqrt{6+...+\sqrt{6+\sqrt{6}}}}}=3\) khi \(n\rightarrow\infty\)

Vậy S < 8

a) Ta có: \(\sqrt{3-2\sqrt{2}}-\sqrt{11+6\sqrt{2}}\)

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

=-4

b) Ta có: \(\sqrt{4-2\sqrt{3}}-\sqrt{7-4\sqrt{3}}+\sqrt{19+8\sqrt{3}}\)

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

\(=3\sqrt{3}+1\)

c) Ta có: \(\sqrt{6-2\sqrt{5}}+\sqrt{9+4\sqrt{5}}-\sqrt{14-6\sqrt{5}}\)

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

\(=3\sqrt{5}-6\)

d) Ta có: \(\sqrt{11-4\sqrt{7}}+\sqrt{23-8\sqrt{7}}+\sqrt{\left(-2\right)^6}\)

\(=\sqrt{7}-2+4-\sqrt{7}+8\)

=10

câu đầu có \(3-12\sqrt{6}< 0\) nên không căn được nên đề bạn sai

\(\sqrt{31-8\sqrt{15}}+\sqrt{24-6\sqrt{15}}\)

\(=\sqrt{4^2-2.4.\sqrt{15}+\left(\sqrt{15}\right)^2}+\sqrt{\left(\sqrt{15}\right)^2-2.\sqrt{15}.3+3^2}\)

\(=\sqrt{\left(4-\sqrt{15}\right)^2}+\sqrt{\left(\sqrt{15}-3\right)^2}=\left|4-\sqrt{15}\right|+\left|\sqrt{15}-3\right|\)

\(=4-\sqrt{15}+\sqrt{15}-3=1\)

\(\sqrt{49-5\sqrt{96}}-\sqrt{49+5\sqrt{96}}=\sqrt{49-20\sqrt{6}}-\sqrt{49+20\sqrt{6}}\)

\(=\sqrt{5^2-2.5.2\sqrt{6}+\left(2\sqrt{6}\right)^2}-\sqrt{5^2+2.5.4\sqrt{6}+\left(2\sqrt{6}\right)^2}\)

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

\(=5-2\sqrt{6}-5-2\sqrt{6}=-4\sqrt{6}\)

\(\sqrt{31-8\sqrt{15}}+\sqrt{24-6\sqrt{15}}\)

\(=4-\sqrt{15}+\sqrt{15}-3\)

=1

Giúp mình với gấp lắm .

\(\dfrac{15}{\sqrt{6}-1}+\dfrac{8}{\sqrt{6}+2}+\dfrac{6}{3-\sqrt{6}}-9\sqrt{6}\)

\(=\dfrac{15\left(\sqrt{6}+1\right)}{\left(\sqrt{6}+1\right)\left(\sqrt{6}-1\right)}+\dfrac{8\left(\sqrt{6}-2\right)}{\left(\sqrt{6}+2\right)\left(\sqrt{6}-2\right)}+\dfrac{6\left(3+\sqrt{6}\right)}{\left(3-\sqrt{6}\right)\left(3+\sqrt{6}\right)}-9\sqrt{6}\)

\(=\dfrac{15\left(\sqrt{6}+1\right)}{6-1}+\dfrac{8\left(\sqrt{6}-2\right)}{6-4}+\dfrac{6\left(3+\sqrt{6}\right)}{9-6}-9\sqrt{6}\)

\(=3\left(\sqrt{6}+1\right)+4\left(\sqrt{6}-2\right)+2\left(3+\sqrt{6}\right)-9\sqrt{6}\)

\(=3\sqrt{6}+3+4\sqrt{6}-8+6+2\sqrt{6}-9\sqrt{6}\)

\(=9\sqrt{6}+1-9\sqrt{6}\)

\(=1\)

\(\sqrt{\left(\sqrt{5}-1\right)\sqrt{14-6\sqrt{5}}}\)

\(=\sqrt{\left(\sqrt{5}-1\right)\sqrt{9-6\sqrt{5}+5}}\)

\(=\sqrt{\left(\sqrt{5}-1\right)\sqrt{3^2-2\cdot3\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}}\)

\(=\sqrt{\left(\sqrt{5}-1\right)\sqrt{\left(3-\sqrt{5}\right)^2}}\)

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

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

\(=\sqrt{3\sqrt{5}-5-3+\sqrt{5}}\)

\(=\sqrt{4\sqrt{5}-8}\)

\(=\sqrt{4\left(\sqrt{5}-2\right)}\)

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

\(\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\)

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

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

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

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

\(=2\sqrt{2}\)

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

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

b: \(=\sqrt{\left(2+\sqrt{3}\right)^2}-\sqrt{5}+2-4+\sqrt{5}\)

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

\(\sqrt{7+4\sqrt{3}}-\sqrt{\left(2-\sqrt{5}\right)^2}-\sqrt{21-8\sqrt{5}}\)

\(=\sqrt{2^2+2\cdot2\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}-\left|2-\sqrt{5}\right|-\sqrt{4^2-2\cdot4\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}\)

\(=\sqrt{\left(2+\sqrt{3}\right)^2}+\left(2-\sqrt{5}\right)-\sqrt{\left(4-\sqrt{5}\right)^2}\)

\(=\left|2+\sqrt{3}\right|+2-\sqrt{5}-\left|4-\sqrt{5}\right|\)

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

\(=\sqrt{3}\)