A family of cosmological solutions with (n+ 1) Ricci-flat spaces in the theory with several scalar fields and multiple exponential potential is obtained when coupling vectors in exponents obey certain relations. Two subclasses of solutions with power-law and exponential behaviour of scale factors are singled out. It is proved that power-law solutions may take place only when coupling vectors are linearly independent and exponential dependence occurs for linearly dependent set of coupling vectors. A subfamily of solutions with accelerated expansion is singled out. A generalized isotropization behaviours of certain classes of general solutions are found. In quantum case exact solutions to Wheeler-DeWitt equation are obtained and special "ground state" wave functions are considered. © SISSA/ISAS 2003.