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=(-1/2) : (-2/3) :( -3/4) :...: (-49/50)
= -1/2 . (-3/2) . (-4/3) . ... . (-50/49)
= -1/2.(-1/2) . (-50)
= - 1/100
\(A = {1\over2}-{3\over4}+{5\over6}-{7\over12}={6\over12}-{9\over12}+{10\over12}-{7\over12}\)\(={0\over12}=0\)
$\textbf{a)}$
$4\cdot\left(\dfrac14\right)^2+25\cdot\left[\left(\dfrac34\right)^3:\left(\dfrac54\right)^3\right]:\left(\dfrac32\right)^3$
$=\dfrac14+25\cdot\left(\dfrac35\right)^3:\left(\dfrac32\right)^3$
$=\dfrac14+25\cdot\dfrac{27}{125}\cdot\dfrac8{27}$
$=\dfrac14+\dfrac85$
$=\dfrac{5+32}{20}$
$=\boxed{\dfrac{37}{20}}.$
$\textbf{b)}$
$2^3+3\cdot\left(\dfrac12\right)^0+(-1)+\left[(-2)^2:\dfrac12\right]-8$
$=8+3\cdot1-1+\dfrac4{1/2}-8$
$=8+3-1+8-8$
$=\boxed{10}.$

P=(1-1/1+2).(1-1/1+2+3)....(1-1/1+2+3+4+...+2011)
=[1-1/(2+1).2:2].[1-1/(3+1).3:2].....[1-1/(2011+1).2011:2]
=(1-2/2.3).(1-2/3.4)...(1-2/2011.2012)
=4/2.3.10/3.4....4046130/2011.2012
=1.4/2.3 .2.5/3.4 ....2010.2013/2011.2012
=1.2....2010/2.3...2011 .4.5....2013/3.4....2012
=1/2011.2013/3
=671/2011