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=\(\frac{1}{\sqrt{7-2\sqrt{6}_{ }}+1}+\frac{1}{\sqrt{7+2\sqrt{6}}+1}\)
=\(\frac{1}{\sqrt{\left(\sqrt{6}-1\right)^2+1}}+\frac{1}{\sqrt{\left(\sqrt{6+1}\right)^2}+1}\)
=\(\frac{1}{\sqrt{6}}+\frac{1}{\sqrt{6}+2}\)
=\(\frac{\sqrt{6}+2+\sqrt{6}}{\sqrt{6}\left(\sqrt{6}+2\right)}\)
=\(\frac{2\sqrt{6}+2}{6+2\sqrt{6}}\)
\(\sqrt{7+\sqrt{24}=\sqrt{7+2\sqrt{6}}=\sqrt{\left(\sqrt{6}+1\right)^2}}\)
ta có :
\(P=\frac{\sqrt{x}+4}{1-7\sqrt{x}}+\frac{\sqrt{x}-2}{\sqrt{x}+1}+\frac{24\sqrt{x}}{\left(\sqrt{x}+1\right)\left(7\sqrt{x}-1\right)}\)
\(\frac{-\left(\sqrt{x}+4\right)\left(\sqrt{x}+1\right)+\left(\sqrt{x}-2\right)\left(7\sqrt{x}-1\right)+24\sqrt{x}}{\left(\sqrt{x}+1\right)\left(7\sqrt{x}-1\right)}=\frac{6x+4\sqrt{x}-2}{\left(\sqrt{x}+1\right)\left(7\sqrt{x}-1\right)}\)
\(=\frac{6\sqrt{x}+2}{7\sqrt{x}-1}\)
Để \(P\ge-6\Leftrightarrow\frac{6\sqrt{x}+2}{7\sqrt{x}-1}\ge-6\Leftrightarrow\frac{48\sqrt{x}-4}{7\sqrt{x}-1}\ge0\)
\(\Leftrightarrow\orbr{\begin{cases}0\le\sqrt{x}\le\frac{1}{12}\\\sqrt{x}>\frac{1}{7}\end{cases}}\Leftrightarrow\orbr{\begin{cases}0\le x\le\frac{1}{144}\\x>\frac{1}{49}\end{cases}}\)
\(=\frac{3\left(\sqrt{x}-1\right)-\left(\sqrt{x}+1\right)-\sqrt{x}+5}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{3\sqrt{x}-3-\sqrt{x}-1-\sqrt{x}+5}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{1}{\sqrt{x}-1}\)
x=\(24-16\sqrt{2}=4^2-2.4.\sqrt{8}+\left(2\sqrt{2}\right)^2=\left(4-2\sqrt{2}\right)^2\)
a) \(P=\frac{3}{\sqrt{x}+1}-\frac{1}{\sqrt{x}-1}-\frac{\sqrt{x}-5}{x-1}\)
\(P=\frac{3\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}-\frac{\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\sqrt{x}-5}{x-1}\)
\(P=\frac{3\sqrt{x}-3-\sqrt{x}-1}{x-1}-\frac{\sqrt{x}-5}{x-1}\)
\(P=\frac{3\sqrt{x}-3-\sqrt{x}-1-\sqrt{x}+5}{x-1}\)
\(P=\frac{\sqrt{x}+1}{x-1}\)
vay \(P=\frac{\sqrt{x}+1}{x-1}\)
b) thay vao P ta duoc:
\(P=\frac{\sqrt{24-16\sqrt{2}}+1}{24-16\sqrt{2}-1}\)
\(P=\frac{\sqrt{\left(2\sqrt{2}\right)^2-2.2.4\sqrt{2}+4^2}+1}{\left(2\sqrt{2}\right)^2-2.2.4\sqrt{2}+4^2-1}\)
\(P=\frac{\sqrt{\left(2\sqrt{2}-4\right)^2}+1}{\left(2\sqrt{2}-4\right)^2-1^2}\)
\(P=\frac{2\sqrt{2}-4+1}{\left(2\sqrt{2}-4-1\right)\left(2\sqrt{2}-4+1\right)}\)
\(P=\frac{2\sqrt{2}-3}{\left(2\sqrt{2}-5\right)\left(2\sqrt{2}-3\right)}\)
\(P=\frac{1}{2\sqrt{2}-5}\)
vay \(P=\frac{1}{2\sqrt{2}-5}\)
\(B=\frac{1}{\sqrt{5}+\sqrt{7}}-\frac{1}{\sqrt{5}-\sqrt{7}}=\frac{\sqrt{5}-\sqrt{7}-\sqrt{5}-\sqrt{7}}{5-7}=\frac{-2\sqrt{7}}{-2}=\sqrt{7}\)
\(C=\sqrt{\frac{4+\sqrt{7}}{4-\sqrt{7}}}+\sqrt{\frac{4-\sqrt{7}}{4+\sqrt{7}}}=\sqrt{\left(\sqrt{\frac{4+\sqrt{7}}{4-\sqrt{7}}}+\sqrt{\frac{4-\sqrt{7}}{4+\sqrt{7}}}\right)^2}\)
\(C=\sqrt{\frac{4+\sqrt{7}}{4-\sqrt{7}}+2\sqrt{\frac{\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)}{\left(4-\sqrt{7}\right)\left(4+\sqrt{7}\right)}}+\frac{4-\sqrt{7}}{4+\sqrt{7}}}\)
\(C=\sqrt{\frac{\left(4+\sqrt{7}\right)^2}{16-7}+\frac{\left(4-\sqrt{7}\right)^2}{16-7}+2}\)
\(C=\sqrt{\frac{\left(4+\sqrt{7}+4-\sqrt{7}\right)^2-2\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)}{16-7}+2}\)
\(C=\sqrt{\frac{16^2-2\left(16-7\right)}{9}+2}=\sqrt{\frac{238}{9}+2}=\sqrt{\frac{256}{9}}=\frac{16}{3}\)
Chúc bạn học tốt ~
\(A=\frac{\sqrt{1}+\sqrt{2}}{1-2}-\frac{\sqrt{2}+\sqrt{3}}{2-3}+\frac{\sqrt{3}+\sqrt{4}}{3-4}-...-\frac{\sqrt{24}+\sqrt{25}}{24-25}\)
\(=-\sqrt{1}-\sqrt{2}+\sqrt{2}+\sqrt{3}-\sqrt{3}-\sqrt{4}+...+\sqrt{24}+\sqrt{25}\)
\(=-\sqrt{1}+\sqrt{25}\)
\(=-1+5\)
\(=4.\)
bạn tách mau ra rồi tính như bình thường thôi mà . bài này dễ chứ ko khó .