Rút gọn biểu thức :
a) \(A=\left(a-1\right)\sqrt{\frac{a}{a-1}}+\sqrt{a\left(a-1\right)}-a\sqrt{\frac{a-1}{a}}\)
b) \(B=\frac{1-\sqrt{x-1}}{\sqrt{x}+2\sqrt{x-1}}-\frac{1+\sqrt{x-1}}{\sqrt{x}-2\sqrt{x-1}}\)
c)\(C=\frac{\sqrt{x-\sqrt{2x-1}}+\sqrt{x+\sqrt{2x-1}}}{\sqrt{x+\sqrt{2x-1}}.\sqrt{x-\sqrt{2x-1}}}.\sqrt{2x-1}\)
d) D=\(\sqrt{\sqrt{a}-\sqrt{\frac{a^2-4}{a}}}-\sqrt{\sqrt{a}+\sqrt{\frac{a^2-4}{a}}}\)
\(A=\left(a-1\right)\sqrt{\frac{a}{a-1}}+\sqrt{a\left(a-1\right)}-a\sqrt{\frac{a-1}{a}}\)
\(A=\sqrt{\left(a-1\right)^2.\frac{a}{a-1}}+\sqrt{a\left(a-1\right)}-\sqrt{a^2.\frac{a-1}{a}}\)
\(A=\sqrt{\left(a-1\right)a}+\sqrt{a\left(a-1\right)}-\sqrt{a\left(a-1\right)}\)
\(A=\sqrt{a\left(a-1\right)}\)
Rút gọn biểu thức:
\(A=\frac{\sqrt{x+2\sqrt{x-1}+\sqrt{x-2\sqrt{x-1}}}}{\sqrt{x+\sqrt{2x-1}+\sqrt{x-\sqrt{2x-1}}}}.\sqrt{2x-1}\)
Rút gọn:
R = \(\left(\frac{2x\sqrt{x}+x-\sqrt{x}}{x\sqrt{x}-1}-\frac{x+\sqrt{x}}{x-1}\right)\cdot\frac{x-1}{2x+\sqrt{x}-1}+\frac{\sqrt{x}}{2\sqrt{x}-1}\)
X = \(\left(\frac{\sqrt{x}+2}{3\sqrt{x}}+\frac{2}{\sqrt{x}+1}-3\right):\frac{2-4\sqrt{x}}{\sqrt{x}+1}-\frac{3\sqrt{x}+1-x}{3\sqrt{x}}\)
Cho:
\(P=\left(\frac{\sqrt{x}+1}{\sqrt{2x}+1}+\frac{\sqrt{2x}+\sqrt{x}}{\sqrt{2x}-1}\right):\left(1+\frac{\sqrt{x}+1}{\sqrt{2x}+1}-\frac{\sqrt{2x}+\sqrt{x}}{\sqrt{2x}-1}\right)\)
a) Rút gọn P
b) Tính P khi \(x=\frac{1}{2}\left(3+2\sqrt{2}\right)\)
Rút gọn: \(A=\left(\frac{1}{\sqrt{x}-\sqrt{x-1}}-\frac{x-3}{\sqrt{x-1}-\sqrt{2}}\right)\left(\frac{2}{\sqrt{2}-\sqrt{x}}-\frac{\sqrt{x}+\sqrt{2}}{\sqrt{2x}-x}\right)\)
Rút gọn
\(1.A=\left(\frac{2\sqrt{x}}{\sqrt{x}+3}+\frac{\sqrt{x}}{\sqrt{x}-3}-\frac{3x+3}{x-9}\right):\left(\frac{2\sqrt{x}-2}{\sqrt{x}-3}-1\right)\)
\(2.B=\left(\frac{\sqrt{a}+1}{\sqrt{ab}+1}+\frac{\sqrt{ab}+\sqrt{a}}{\sqrt{ab}-1}-1\right):\left(\frac{\sqrt{a}+1}{\sqrt{ab}+1}-\frac{\sqrt{ab}+\sqrt{a}}{\sqrt{ab}-1}+1\right)\)
\(3.C=\left(\frac{2x-1+\sqrt{x}}{1-x}+\frac{2x\sqrt{x}+x-\sqrt{x}}{1+x\sqrt{x}}\right).\left(\frac{\left(x-\sqrt{x}\right)\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}\right)\)
\(\frac{\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}}{\sqrt{x+\sqrt{2x-1}}-\sqrt{x-\sqrt{2x-1}}}\)
rút gọn biểu thức
\(\frac{A}{\sqrt{2}}\)=\(\frac{\sqrt{x-1+2\sqrt{x-1}+1}+\sqrt{x-1-2\sqrt{x-1}+1}}{\sqrt{2x-1+2\sqrt{2x-1}+1}-\sqrt{2x-1-2\sqrt{2x-1}+1}}\) (DK \(x\ge1\)
\(=\frac{\sqrt{x-1}+1+\left|\sqrt{x-1}-1\right|}{\sqrt{2x-1}+1-\left|\sqrt{2x-1}-1\right|}\)
vs \(x\ge2\) \(\frac{\sqrt{x-1}+1+\sqrt{x-1}-1}{\sqrt{2x-1}+1-\sqrt{2x-1}+1}=\frac{2\sqrt{x-1}}{2}=\sqrt{x-1}\) \(\Rightarrow A=\sqrt{2x-2}\)
vs \(1\le x< 2\) \(\frac{\sqrt{x-1}+1+1-\sqrt{x-1}}{\sqrt{2x-1}+1-1+\sqrt{2x-1}}=\frac{1}{\sqrt{2x-1}}\) \(\Rightarrow A=\frac{\sqrt{2}}{\sqrt{2x-1}}\)
\(\sqrt{2X-1}\ge1\Leftrightarrow X\ge1\)NEN SUY RA THEO CACH LAM CUA TO
THOI U AM BUSY SEE YOU AGAIN
làm zì mà dài vậy \(\sqrt{x+2\sqrt{x-1}}=\sqrt{x-1+2\sqrt{x-1}+1}=\sqrt{\left(\sqrt{x-1}-1\right)^2}\)
rút gọn: \(P=\left(\frac{2x\sqrt{x}+x-\sqrt{x}}{x\sqrt{x}-1}-\frac{x+\sqrt{x}}{x-1}\right)\times\frac{x-1}{2x+\sqrt{x}-1}+\frac{\sqrt{x}}{2\sqrt{x}-1}\)
RÚT GỌN
\(A=\frac{\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}}{\sqrt{x+\sqrt{2x-1}}-\sqrt{x-\sqrt{2x-1}}}:\sqrt{x-1}\)(VỚI x>2)
\(\frac{1}{\sqrt{2}}.A=\frac{\sqrt{\left(x-1\right)+2\sqrt{x-1}+1}+\sqrt{\left(x-1\right)-2\sqrt{x-1}+1}}{\sqrt{\left(2x-1\right)+2\sqrt{2x-1}+1}-\sqrt{\left(2x-1\right)-2\sqrt{2x-1}+1}}\)
\(=\frac{\sqrt{\left[\left(\sqrt{x-1}+1\right)\right]^2}+\sqrt{\left[\left(\sqrt{x-1}-1\right)^2\right]}}{\sqrt{\left[\sqrt{2x-1}+1\right]^2}-\sqrt{\left[\left(\sqrt{2x-1}\right)-1\right]^2}}\)
\(=\frac{\left|\sqrt{x-1}+1\right|+\left|\sqrt{x-1}-1\right|}{\left|\sqrt{2x-1}+1\right|-\left|\sqrt{2x-1}-1\right|}\)
DO X>2 NÊN TOÀN BỘ BIỂU THỨC TRONG TRỊ TUYỆT ĐỐI ĐỀU DƯƠNG
\(\frac{1}{\sqrt{2}}.A=\frac{2\sqrt{x-1}}{2}=\sqrt{x-1}\)
=>\(A=\frac{\sqrt{x-1}}{\sqrt{2}}\)
