Answer
$$\left( a \right)4,\,\,\,\,\left( b \right)4,\,\,\,\left( c \right)\frac{1}{4}$$
Work Step by Step
$$\eqalign{
& \left( {\bf{a}} \right)\,\,\,\, - {8^{2/3}} \cr
& {\text{Use the property }}{\left( {{a^m}} \right)^n} = {a^{mn}} \cr
& = - {\left( {{8^{1/3}}} \right)^2} \cr
& {\text{Use the property }}{a^{m/n}} = \root n \of {{a^m}} \cr
& = - {\left( {\root 3 \of 8 } \right)^2} \cr
& = - {\left( 2 \right)^2} \cr
& {\text{simplify}} \cr
& = - 4 \cr
& \cr
& \left( {\bf{b}} \right)\,\,\,\,{\left( { - 8} \right)^{2/3}} \cr
& {\text{Use the property }}{\left( {{a^m}} \right)^n} = {a^{mn}} \cr
& = {\left( { - {8^{1/3}}} \right)^2} \cr
& {\text{Use the property }}{a^{m/n}} = \root n \of {{a^m}} \cr
& = {\left( {\root 3 \of { - 8} } \right)^2} \cr
& = {\left( { - 2} \right)^2} \cr
& {\text{simplify}} \cr
& = 4 \cr
& \cr
& \left( {\bf{c}} \right)\,\,\,\,{8^{ - 2/3}} \cr
& {\text{Use the property }}{\left( {{a^m}} \right)^n} = {a^{mn}} \cr
& = {\left( {{8^{1/3}}} \right)^{ - 2}} \cr
& {\text{Use the property }}{a^{m/n}} = \root n \of {{a^m}} \cr
& = {\left( {\root 3 \of 8 } \right)^{ - 2}} \cr
& = {\left( 2 \right)^{ - 2}} \cr
& {\text{Use the property }}{a^{ - n}} = \frac{1}{{{a^n}}} \cr
& = \frac{1}{{{{\left( 2 \right)}^2}}} \cr
& {\text{simplify}} \cr
& = \frac{1}{4} \cr} $$