Rút gọn biểu thức:
\(\sqrt{x+\sqrt{x^2-4}}-4\sqrt{x-\sqrt{x^2-4}}\)
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25 tháng 7 2021
\(A=\left(\sqrt{x}+\dfrac{4\sqrt{x}}{\sqrt{x}-2}\right):\left(\dfrac{\sqrt{x}}{\sqrt{x}-2}+\dfrac{4}{2\sqrt{x}-x}\right)\)ĐK : x > 0 ; x \(\ne\)4
\(=\left(\dfrac{x+2\sqrt{x}}{\sqrt{x}-2}\right):\left(\dfrac{x-4}{\sqrt{x}\left(\sqrt{x}-2\right)}\right)=\dfrac{x\left(x-4\right)}{\left(\sqrt{x}-2\right)\left(x-4\right)}\)
\(=\dfrac{x}{\sqrt{x}-2}\)
ND
1
AT
1 tháng 7 2021
ĐKXĐ: \(x\ge2\)
\(A=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}\)
\(=\sqrt{x-2+2.\sqrt{x-2}.\sqrt{2}+2}+\sqrt{x-2-2.\sqrt{x-2}.\sqrt{2}+2}\)
\(=\sqrt{\left(\sqrt{x-2}+\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{x-2}-\sqrt{2}\right)^2}\)
\(=\left|\sqrt{x-2}+\sqrt{2}\right|+\left|\sqrt{x-2}-\sqrt{2}\right|=\sqrt{x-2}+\sqrt{2}+\left|\sqrt{x-2}-\sqrt{2}\right|\)
Xét \(x\ge4\Rightarrow\sqrt{x-2}\ge\sqrt{2}\)
\(\Rightarrow A=\sqrt{x-2}+\sqrt{2}+\sqrt{x-2}-\sqrt{2}=2\sqrt{x-2}\)
Xét \(0\le x< 4\Rightarrow\sqrt{x-2}< \sqrt{2}\)
\(\Rightarrow A=\sqrt{x-2}+\sqrt{2}-\sqrt{x-2}+\sqrt{2}=2\sqrt{2}\)
N
23 tháng 11 2021
\(\dfrac{2\left(2-\sqrt{x}\right)}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}+\dfrac{2\sqrt{x}\left(2+\sqrt{x}\right)}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}-\dfrac{2x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}=\dfrac{4-2\sqrt{x}+4\sqrt{x}+2x-2x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}=\dfrac{4-2\sqrt{x}}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}=\dfrac{2\left(2-\sqrt{x}\right)}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}=\dfrac{2}{2+\sqrt{x}}\)
18 tháng 2 2022
đk : x >= 0, x khác 4
\(=\dfrac{x+2\sqrt{x}-\left(x-\sqrt{x}-2\right)-\sqrt{x}-4}{x-4}\)
\(=\dfrac{2\sqrt{x}-2}{x-4}=\dfrac{2\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
10 tháng 11 2021
a.\(A=\dfrac{x^2-4x+4}{x^3-2x^2-\left(4x-8\right)}=\dfrac{\left(x-2\right)^2}{x^2\left(x-2\right)-4\left(x-2\right)}=\dfrac{\left(x-2\right)^2}{\left(x^2-4\right)\left(x-2\right)}=\dfrac{x-2}{\left(x-2\right)\left(x+2\right)}=\dfrac{1}{x+2}\)
10 tháng 11 2021
\(A=\dfrac{\left(x-2\right)^2}{x^2\left(x-2\right)-4\left(x-2\right)}\left(x\ne\pm2\right)\\ A=\dfrac{\left(x-2\right)^2}{\left(x-2\right)^2\left(x+2\right)}=\dfrac{1}{x+2}\\ B=\dfrac{x+2-x+\sqrt{x}-1}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\cdot\dfrac{4\sqrt{x}}{3}\left(x>0\right)\\ B=\dfrac{4\sqrt{x}\left(\sqrt{x}+1\right)}{3\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}=\dfrac{4\sqrt{x}}{3\left(x-\sqrt{x}+1\right)}\)
20 tháng 12 2022
\(=\dfrac{x-4\sqrt{x}+4\sqrt{x}+16}{x-16}\cdot\dfrac{\sqrt{x}+2}{\sqrt{x}+16}=\dfrac{\left(x+16\right)\left(\sqrt{x}+2\right)}{\left(x-16\right)\left(\sqrt{x}+16\right)}\)
26 tháng 8 2021
\(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}+\dfrac{2\sqrt{x}}{\sqrt{x}+2}+\dfrac{2+5\sqrt{x}}{4-x}=\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)+2\sqrt{x}\left(\sqrt{x}-2\right)-2-5\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{x+3\sqrt{x}+2+2x-4\sqrt{x}-2-5\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{3x-6\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{3\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{3\sqrt{x}}{\sqrt{x}+2}\)
26 tháng 8 2021
Ta có: \(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}+\dfrac{2\sqrt{x}}{\sqrt{x}+2}+\dfrac{5\sqrt{x}+2}{4-x}\)
\(=\dfrac{x+3\sqrt{x}+2+2x-4\sqrt{x}-5\sqrt{x}-2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{3x-6\sqrt{x}}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{3\sqrt{x}}{\sqrt{x}+2}\)
Đặt\(A=\sqrt{x+\sqrt{x^2-4}}-4.\sqrt{x-\sqrt{x^2-4}}\)
\(A^2=x+\sqrt{x^2-4}+16.\left(x-\sqrt{x^2-4}\right)-2.4.\sqrt{x^2-\left(\sqrt{x^2-4}\right)^2}\)
\(A^2=x+\sqrt{x^2-4}+16x-16.\sqrt{x^2-4}-8.\sqrt{x^2-x^2+4}\)
\(A^2=17x-15.\sqrt{x^2-4}-16\)
mình làm đến đây đc thôi, sorry
Dễ thây \(x\ge2\)
\(A=\sqrt{x+\sqrt{x^2-4}}-4\sqrt{x-\sqrt{x^2-4}}\)
\(=\sqrt{\frac{2x+2\sqrt{\left(x+2\right)\left(x-2\right)}}{2}}-4\sqrt{\frac{2x-2\sqrt{\left(x+2\right)\left(x-2\right)}}{2}}\)
\(=\sqrt{\frac{\left(x+2\right)+2\sqrt{\left(x+2\right)\left(x-2\right)}+\left(x-2\right)}{2}}-4\sqrt{\frac{\left(x+2\right)-2\sqrt{\left(x+2\right)\left(x-2\right)}+\left(x-2\right)}{2}}\)
\(=\sqrt{\frac{\left(\sqrt{\left(x+2\right)}+\sqrt{\left(x-2\right)}\right)^2}{2}}-4\sqrt{\frac{\left(\sqrt{\left(x+2\right)}-\sqrt{\left(x-2\right)}\right)^2}{2}}\)
\(=\frac{1}{\sqrt{2}}\left[\left(\sqrt{x+2}+\sqrt{x-2}\right)-4\left(\sqrt{x+2}-\sqrt{x-2}\right)\right]\)
\(=\frac{1}{\sqrt{2}}\left(-3\sqrt{x+2}+5\sqrt{x-2}\right)\)