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a) N = \(\frac{x}{x-4}+\frac{1}{\sqrt{x}-2}+\frac{1}{\sqrt{x}+2}\)
N = \(\frac{x+\sqrt{x}+2+\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
N = \(\frac{x+2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
N = \(\frac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
N = \(\frac{\sqrt{x}}{\sqrt{x}-2}\)
b) Với x \(\ge\)0; x \(\ne\)4
Ta có: N = \(\frac{1}{-3}\) <=> \(\frac{\sqrt{x}}{\sqrt{x}-2}=\frac{1}{-3}\)
=> \(-3\sqrt{x}=\sqrt{x}-2\)
<=> \(-4\sqrt{x}=-2\)
<=> \(\sqrt{x}=\frac{1}{2}\)
<=> \(x=\frac{1}{4}\)
c) x = 25 => N = \(\frac{\sqrt{25}}{\sqrt{25}-2}=\frac{5}{5-3}=\frac{5}{2}\)
a) \(N=\frac{x}{x-4}+\frac{1}{\sqrt{x}-2}+\frac{1}{\sqrt{x}+2}\)
\(N=\frac{x+\sqrt{x}+2+\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(N=\frac{x+2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(N=\frac{\left(\sqrt{x}+2\right)\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(N=\frac{\sqrt{x}}{\sqrt{x}-2}\)
b) \(N=-\frac{1}{3}\)
\(\Leftrightarrow\frac{\sqrt{x}}{\sqrt{x}-2}=-\frac{1}{3}\)
\(\Leftrightarrow3\sqrt{x}=2-\sqrt{x}\)
\(\Leftrightarrow4\sqrt{x}=2\)
\(\Leftrightarrow\sqrt{x}=\frac{1}{2}\Rightarrow x=\frac{1}{4}\)
c) \(N=\frac{\sqrt{25}}{\sqrt{25}-2}=\frac{5}{5-2}=\frac{5}{3}\)
a) 10n + 1 - 6.10n
= 10n . 10 - 6 . 10n
= 10n . (10 - 6)
= 10n . 4
b) 2n + 3 + 2n + 2 - 2n + 1 + 2n
= 2n . 23 + 2n . 22 - 2n . 2 + 2n . 1
= 2n . (8 + 4 - 2 + 1)
= 2n . 11
\(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+......+\frac{1}{\sqrt{n-1}+\sqrt{n}}=\frac{\sqrt{1}-\sqrt{2}}{\left(\sqrt{1}+\sqrt{2}\right)\left(\sqrt{1}-\sqrt{2}\right)}+\frac{\sqrt{2}-\sqrt{3}}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}+......+\frac{\sqrt{n-1}-\sqrt{n}}{\left(\sqrt{n-1}+\sqrt{n}\right)\left(\sqrt{n-1}-\sqrt{n}\right)}\)\(=\frac{\sqrt{1}-\sqrt{2}}{1-2}+\frac{\sqrt{2}-\sqrt{3}}{2-3}+......+\frac{\sqrt{n-1}-\sqrt{n}}{n-1-n}\)
=\(-\left(\sqrt{1}-\sqrt{2}+\sqrt{2}-\sqrt{3}+......+\sqrt{n-1}-\sqrt{n}\right)=-\left(1-\sqrt{n}\right)=\sqrt{n}-1\)
Với n\(\in N\)* có: \(\dfrac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\dfrac{1}{\sqrt{n\left(n+1\right)}\left(\sqrt{n+1}+\sqrt{n}\right)}\)\(=\dfrac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n\left(n+1\right)}\left(n+1-n\right)}=\dfrac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n\left(n+1\right)}}\)\(=\dfrac{1}{\sqrt{n}}-\dfrac{1}{\sqrt{n+1}}\)
\(\Rightarrow\)\(\dfrac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\dfrac{1}{\sqrt{n}}-\dfrac{1}{\sqrt{n+1}}\) (*)
a) Áp dụng (*) vào T
\(\Rightarrow T=1-\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{99}}-\dfrac{1}{\sqrt{100}}\)\(=1-\dfrac{1}{10}=\dfrac{9}{10}\)
b) Có \(VT=1-\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{n}}-\dfrac{1}{\sqrt{n+1}}\)\(=1-\dfrac{1}{\sqrt{n+1}}=\dfrac{4}{5}\)
\(\Leftrightarrow\sqrt{n+1}=5\Leftrightarrow n=24\) (tm)
Vậy n=24.
a: \(P=\dfrac{\sqrt{3}\left(2+\sqrt{3}\right)}{\sqrt{3}}+\dfrac{\sqrt{2}\left(\sqrt{2}-1\right)}{1}-\sqrt{3}-\sqrt{2}\)
\(=2+\sqrt{3}+2-\sqrt{2}-\sqrt{3}-\sqrt{2}\)
\(=4-2\sqrt{2}\)
b: \(N=\left(1-\dfrac{\sqrt{5}\left(\sqrt{5}+1\right)}{\sqrt{5}+1}\right)\left(\dfrac{-\sqrt{5}\left(1-\sqrt{5}\right)}{1-\sqrt{5}}-1\right)\)
\(=\left(1-\sqrt{5}\right)\left(-\sqrt{5}-1\right)\)
\(=\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)=5-1=4\)
Chọn đáp án C.
Ta có: