Cho \(B=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

B² = \(4+\sqrt{10+2\sqrt{5}}+2\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\left(4-\sqrt{10+2\sqrt{5}}\right)}+4-\sqrt{10+2\sqrt{5}}\)

= \(8+2\sqrt{16-\left(10+2\sqrt{5}\right)}\)

= \(8+2\sqrt{6-2\sqrt{5}}\)

= \(8+2\sqrt{5-2\sqrt{5}+1}\)

= \(8+2\sqrt{\left(\sqrt{5}-1\right)^2}\)

= \(8+2.\left(\sqrt{5}-1\right)\) (do \(\sqrt{5}>1\))

= \(6+2\sqrt{5}\)

= \(5+2\sqrt{5}+1\)

= \(\left(\sqrt{5}+1\right)^2\)

=> B = \(\sqrt{5}+1\)

Đặt \(A=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

\(\Rightarrow A^2=\left(\sqrt{4+\sqrt{10+2\sqrt{5}}}\right)^2+\left(\sqrt{4-\sqrt{10+2\sqrt{5}}}\right)^2+2\sqrt{4+\sqrt{10+2\sqrt{5}}}\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

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

\(=8+2\sqrt{16-\left(10+2\sqrt{5}\right)}=8+2\sqrt{6-2\sqrt{5}}\)

\(=8+2\sqrt{\left(\sqrt{5}\right)^2-2.\sqrt{5}.1+1^2}=8+2\sqrt{\left(\sqrt{5}-1\right)^2}\)

\(8+2\left|\sqrt{5}-1\right|=8+2\left(\sqrt{5}-1\right)=6+2\sqrt{5}=\left(\sqrt{5}\right)^2+2.\sqrt{5}.1+1^2\)

\(=\left(\sqrt{5}+1\right)^2\Rightarrow A=\sqrt{5}+1\left(A>0\right)\)

Đặt \(A=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

\(A^2=4+\sqrt{10+2\sqrt{5}}+4-\sqrt{10+2\sqrt{5}}+2\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\left(4-\sqrt{10+2\sqrt{5}}\right)}\\ A^2=8+2\sqrt{16-10-2\sqrt{5}}=8+2\sqrt{6-2\sqrt{5}}\\ A^2=8+2\left(\sqrt{5}-1\right)=6+2\sqrt{5}=\left(\sqrt{5}+1\right)^2\\ A=\sqrt{5}+1\)

Đặt \(A=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

\(\Leftrightarrow A^2=4+\sqrt{10+2\sqrt{5}}+4-\sqrt{10+2\sqrt{5}}+2\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\left(4-\sqrt{10+2\sqrt{5}}\right)}\)

\(\Leftrightarrow A^2=8+2\sqrt{16-10-2\sqrt{5}}\\ \Leftrightarrow A^2=8+2\sqrt{6-2\sqrt{5}}\\ \Leftrightarrow A^2=8+2\left(\sqrt{5}-1\right)\\ \Leftrightarrow A^2=6+2\sqrt{5}=\left(\sqrt{5}+1\right)^2\\ \Leftrightarrow A=\sqrt{5}+1\)

Vậy \(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}=\sqrt{5}+1\)

Trả lời:

\(A=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

\(A^2=\left(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\right)^2\)

\(A^2=4+\sqrt{10+2\sqrt{5}}+2.\sqrt{4+\sqrt{10+2\sqrt{5}}}.\sqrt{4-\sqrt{10+2\sqrt{5}}}+4-\sqrt{10+2\sqrt{5}}\)

\(A^2=8+2\sqrt{16-10-2\sqrt{5}}\)

\(A^2=8+2\sqrt{6-2\sqrt{5}}\)

\(A^2=8+2\sqrt{5-2\sqrt{5}+1}\)

\(A^2=8+2\sqrt{\left(\sqrt{5}-1\right)^2}\)

\(A^2=8+2.\left(\sqrt{5}+1\right)\)

\(A^2=8+2\sqrt{5}-2\)

\(A^2=6+2\sqrt{5}\)

\(A^2=5+2\sqrt{5}+1\)

\(A^2=\left(\sqrt{5}+1\right)^2\)

\(A=\sqrt{5}+1\)

\(B=\sqrt{4+\sqrt{15}}+\sqrt{4-\sqrt{15}}-2\sqrt{3-\sqrt{5}}\)

\(\sqrt{2}B=\sqrt{2}\sqrt{4+\sqrt{15}}+\sqrt{2}\sqrt{4-\sqrt{15}}-\sqrt{2}.2\sqrt{3-\sqrt{5}}\)

\(\sqrt{2}B=\sqrt{8+2\sqrt{15}}+\sqrt{8-2\sqrt{15}}-2\sqrt{6-2\sqrt{5}}\)

\(\sqrt{2}B=\sqrt{5+2\sqrt{15}+3}+\sqrt{5-2\sqrt{15}+3}-2\sqrt{5-2\sqrt{5}+1}\)

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

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

\(\sqrt{2}B=2\)

\(B=\sqrt{2}\)

Cảm ơn bạn nhiều nha UvU 

\(A^2=4+\sqrt{10+2\sqrt{5}}+2.\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right).\left(4-\sqrt{10+2\sqrt{5}}\right)}+4-\sqrt{10+2\sqrt{5}}\)

\(A^2=8+2.\sqrt{4^2-\left(10+2\sqrt{5}\right)}=8+2.\sqrt{6-2\sqrt{5}}\)

\(A^2=8+2.\sqrt{5-2\sqrt{5}.1+1}=8+2.\sqrt{\left(\sqrt{5}-1\right)^2}=8+2.\left(\sqrt{5}-1\right)\)

\(A^2=6+2\sqrt{5}=5+2\sqrt{5}+1=\left(\sqrt{5}+1\right)^2\)

=> \(A=\sqrt{5}+1\) (Do A > 0)

Bạn xem lại đề

1.

a, \(2\sqrt{18}-4\sqrt{50}-3\sqrt{32}=6\sqrt{2}-20\sqrt{2}-12\sqrt{2}=-2\sqrt{2}\)

b, \(\sqrt{14-6\sqrt{5}}+\sqrt{14+6\sqrt{5}}=\sqrt{\left(\sqrt{5}-3\right)^2}+\sqrt{\left(\sqrt{5}+3\right)^2}\)

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

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

c, \(\dfrac{\sqrt{10}+10}{1+\sqrt{10}}-\dfrac{5\sqrt{2}-2\sqrt{5}}{\sqrt{5}-\sqrt{2}}=\dfrac{\sqrt{10}\left(1+\sqrt{10}\right)}{1+\sqrt{10}}-\dfrac{\sqrt{10}\left(\sqrt{5}-\sqrt{2}\right)}{\sqrt{5}-\sqrt{2}}\)

\(=\sqrt{10}-\sqrt{10}=0\)

2.

ĐK: \(x\in R\)

\(\sqrt{9x^2-30x+25}=5\)

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

\(\Leftrightarrow\left|3x-5\right|=5\)

\(\Leftrightarrow\left[{}\begin{matrix}3x-5=5\\3x-5=-5\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{10}{3}\\x=0\end{matrix}\right.\)

Vậy ...