$\begin{array}{l}
a)\,\,{\left( {x + y} \right)^2} - {\left( {x - y} \right)^2}\\
= {x^2} + 2xy + {y^2} - \left( {{x^2} - 2xy + {y^2}} \right)\\
= {x^2} + 2xy + {y^2} - {x^2} + 2xy - {y^2}\\
= 4xy\\
b)\,\,{\left( {a + b} \right)^3} + {\left( {a - b} \right)^3} - 2{a^3}\\
= {a^3} + 3{a^2}b + 3a{b^2} + {b^3} + \left( {{a^3} - 3{a^2}b + 3a{b^2} - {b^3}} \right) - 2{a^3}\\
= {a^3} + 3{a^2}b + 3a{b^2} + {b^3} + {a^3} - 3{a^2}b + 3a{b^2} - {b^3} - 2{a^3}\\
= 6a{b^2}\\
c)\,{9^8}{.2^8} - \left( {{{18}^4} - 1} \right)\left( {{{18}^4} + 1} \right)\\
= \,{\left( {9.2} \right)^8} - \left[ {{{\left( {{{18}^4}} \right)}^2} - {1^2}} \right]\\
= {18^8} - \left( {{{18}^8} - 1} \right) = {18^8} - {18^8} + 1 = 1
\end{array}$
