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$\textbf{a)}$
$4\cdot\left(\dfrac14\right)^2+25\cdot\left[\left(\dfrac34\right)^3:\left(\dfrac54\right)^3\right]:\left(\dfrac32\right)^3$
$=\dfrac14+25\cdot\left(\dfrac35\right)^3:\left(\dfrac32\right)^3$
$=\dfrac14+25\cdot\dfrac{27}{125}\cdot\dfrac8{27}$
$=\dfrac14+\dfrac85$
$=\dfrac{5+32}{20}$
$=\boxed{\dfrac{37}{20}}.$
$\textbf{b)}$
$2^3+3\cdot\left(\dfrac12\right)^0+(-1)+\left[(-2)^2:\dfrac12\right]-8$
$=8+3\cdot1-1+\dfrac4{1/2}-8$
$=8+3-1+8-8$
$=\boxed{10}.$
\(\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{n+1}\right)\)
\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{n}{n+1}\)
\(=\frac{1}{n+1}\)
\(1+\frac{1}{2}.\left(1+2\right)+\frac{1}{3}.\left(1+2+3\right)...+\frac{1}{20}.\left(1+2+3+...+20\right)\)
\(=1+\frac{1}{2}.2.3:2+\frac{1}{3}.3.4:2+\frac{1}{4}.4.5:2+...+\frac{1}{20}.20.21:2\)
\(=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+\frac{5}{2}+...+\frac{21}{2}\)
\(=\frac{2+3+4+5+...+21}{2}=115\)
\(-4\frac{1}{3}.\left(\frac{1}{2}-\frac{1}{6}\right)\le x\le-\frac{2}{3}.\left(\frac{1}{3}-\frac{1}{2}-\frac{3}{4}\right)\)
\(\Rightarrow-\frac{13}{3}.\left(\frac{3}{6}-\frac{1}{6}\right)\le x\le-\frac{2}{3}.\left(\frac{4}{12}-\frac{6}{12}-\frac{9}{12}\right)\)
\(\Rightarrow-\frac{13}{3}.\frac{2}{6}\le x\le-\frac{2}{3}.\frac{-11}{12}\)
\(\Rightarrow\frac{-13}{9}\le x\le\frac{11}{18}\)
\(\Rightarrow\frac{-26}{18}\le x\le\frac{11}{18}\)
=> -1,44444444444........... ≤ x ≤ 0,6111111111...........
Mà x ∈ Z
=> x ∈ { -1 ; 0 }