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\(a,\frac{5}{6}-2\sqrt{\frac{4}{9}}+\sqrt{\left(-2\right)^2}\)
\(=\frac{5}{6}-2.\frac{2}{3}+2\)
\(=\frac{5}{6}-\frac{4}{6}+\frac{12}{6}\)
\(=\frac{5-4+12}{6}=\frac{13}{6}\)
\(b,\left(-3\right)^2.\left(\frac{1}{3}\right)^3:\left[\left(-\frac{2}{3}\right)^3-1\frac{1}{3}\right]-\left(-200\right)^0\)
\(=9.\frac{1}{27}:\left(-\frac{8}{27}-\frac{5}{3}\right)-1\)
\(=\frac{1}{3}:\left(-\frac{8}{27}-\frac{45}{27}\right)-1\)
\(=\frac{1}{3}:\left(-\frac{53}{27}\right)-1\)
\(=\frac{1}{3}.\left(-\frac{27}{53}\right)-1\)
\(=-\frac{9}{53}-1=-\frac{9}{53}-\frac{53}{53}\)
\(=-\frac{62}{53}\)
\(c,\left(-0,5-\frac{3}{5}\right):\left(-3\right)+\frac{1}{3}-\left(-\frac{1}{6}\right):2\)
\(=\left(-\frac{1}{2}-\frac{3}{5}\right).\frac{1}{3}+\frac{1}{3}-\left(-\frac{1}{6}\right).\left(-\frac{1}{2}\right)\)
\(=\left(-\frac{5}{10}-\frac{6}{10}\right).\frac{1}{3}+\frac{1}{3}-\frac{1}{12}\)
\(=-\frac{11}{10}.\frac{1}{3}+\frac{1}{3}-\frac{1}{12}\)
\(=\frac{1}{3}\left(-\frac{11}{10}-\frac{1}{12}\right)\)
\(=\frac{1}{3}\left(-\frac{66}{60}-\frac{5}{60}\right)\)
\(=\frac{1}{3}.\left(-\frac{71}{60}\right)\)
\(=-\frac{71}{180}\)
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d)s=8+7^2+7^3+...+7^99=1+7+7^2+7^3+...+7^99
7s=7+7^2+7^3+7^4+...+7^100
7s-s=7^100-1
s=(7^100-1)/6
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\(=\dfrac{101}{2}\left(4+\dfrac{5}{3}-2-\dfrac{5}{3}\right)=\dfrac{101}{2}\cdot2=101\)
1:
i: \(=\dfrac{15}{34}+\dfrac{19}{34}+\dfrac{7}{21}+\dfrac{2}{3}-1-\dfrac{15}{37}\)
\(=1+\dfrac{1}{3}+\dfrac{2}{3}-1-\dfrac{15}{37}\)
\(=1-\dfrac{15}{37}=\dfrac{22}{37}\)
j: \(=1-\left(-27\right)+\dfrac{1}{2}:\dfrac{-1}{8}\)
\(=1+27-4=24\)
k: \(=-8+\dfrac{1}{2}\cdot\dfrac{-4}{1}-15\)
\(=-8-2-15=-25\)
l: \(=3:\dfrac{9}{4}+\dfrac{1}{9}\cdot6\)
\(=3\cdot\dfrac{4}{9}+\dfrac{1}{9}\cdot6\)
\(=\dfrac{4}{3}+\dfrac{2}{3}=2\)
m: \(=9\cdot\dfrac{1}{3}-\left(-27\right)=3+27=30\)
n: \(\sqrt{\dfrac{16}{25}}\cdot\sqrt{\dfrac{121}{64}}-1\dfrac{3}{10}\)
\(=\dfrac{4}{5}\cdot\dfrac{11}{8}-\dfrac{13}{10}\)
\(=\dfrac{11}{10}-\dfrac{13}{10}=-\dfrac{2}{10}=-\dfrac{1}{5}\)
o: \(=\dfrac{9}{8}\cdot12-\dfrac{1}{2}\)
\(=\dfrac{27}{2}-\dfrac{1}{2}=\dfrac{26}{2}=13\)
p: \(=\dfrac{3^2\cdot2^2+3^2\cdot3\cdot2^2+3^2}{-13}\)
\(=\dfrac{3^2\left(2^2+3\cdot2^2+3^2\right)}{-13}\)
\(=\dfrac{9\cdot\left(4+3\cdot4+9\right)}{-13}\)
\(=\dfrac{9\cdot25}{-13}=-\dfrac{225}{13}\)