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\(A=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{99}+\sqrt{100}}\)
=\(\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{100}-\sqrt{99}\)
=10-1=9
\(\frac{1}{\sqrt{n}+\sqrt{n+1}}=\sqrt{n+1}-\sqrt{n}
\)
r thay n là lm đk
\(P=\frac{\sqrt{a}\left(16-\sqrt{a}\right)}{a-4}+\frac{3+2\sqrt{a}}{2-\sqrt{a}}-\frac{2-3\sqrt{a}}{\sqrt{a+2}}\)
\(=\frac{\sqrt{a}\left(16-\sqrt{a}\right)}{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}-\frac{3+2\sqrt{a}}{\sqrt{a}-2}-\frac{2-3\sqrt{a}}{\sqrt{a}+2}\)
\(=\frac{\sqrt{a}\left(16-\sqrt{a}\right)-\left(3+2\sqrt{a}\right)\left(\sqrt{a}+2\right)-\left(2-3\sqrt{a}\right)\left(\sqrt{a}-2\right)}{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}\)
\(=\frac{16\sqrt{a}-a-3\sqrt{a}-6-2a-4\sqrt{a}-2\sqrt{a}+4+3a-6\sqrt{a}}{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}\)
\(=\frac{\sqrt{a}-2}{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}\)
\(=\frac{1}{\sqrt{a}+2}\)
b,Với ĐKXĐ,ta có: \(P=\frac{1}{\sqrt{a}-2}\)
Để P = 1/2
thì: \(\frac{1}{\sqrt{a}-2}=\frac{1}{2}\)
\(\Leftrightarrow\sqrt{a}-2=2\)
\(\Leftrightarrow\sqrt{a}=4\)
\(\Leftrightarrow a=16\left(tm\right)\)
Q = \(\frac{\sqrt{a}+3}{\sqrt{a}-2}\)- \(\frac{\sqrt{a}-1}{\sqrt{a}+2}\)+ \(\frac{4-4\sqrt{a}}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}\)
= \(\frac{\left(\sqrt{a}+3\right)\left(\sqrt{a}+2\right)-\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)+4-4\sqrt{a}}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}\)
=\(\frac{a+5\sqrt{a}+6-a+3\sqrt{a}-2+4-4\sqrt{a}}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}\)
= \(\frac{8+4\sqrt{a}}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}\)
= \(\frac{4\left(\sqrt{a}+2\right)}{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}\)
= \(\frac{4}{\sqrt{a}-2}\)
\(Q=\frac{\sqrt{a+3}}{\sqrt{a-2}}-\frac{\sqrt{a-1}}{\sqrt{a+2}}+\frac{4-4\sqrt{a}}{\left(\sqrt{a-2}\right)\left(\sqrt{a+2}\right)}\)
\(Q=\frac{\left(\sqrt{a+3}\right)\left(\sqrt{a+2}\right)-\left(\sqrt{a-1}\right)\left(\sqrt{a-2}\right)+4-4\sqrt{a}}{\left(\sqrt{a-2}\right)\left(\sqrt{a+2}\right)}\)
\(Q=\frac{a+5\sqrt{a}+6-a+3\sqrt{a-2}+4-4\sqrt{a}}{\left(\sqrt{a-2}\right)\left(\sqrt{a+2}\right)}\)
\(Q=\frac{8+4\sqrt{a}}{\left(\sqrt{a-2}\right)\left(\sqrt{a+2}\right)}\)
\(Q=\frac{4\left(\sqrt{a+2}\right)}{\left(\sqrt{a+2}\right)\left(\sqrt{a-2}\right)}\)
\(Q=\frac{4}{\sqrt{a-2}}\)