Yes you can. The DHCP server is still running in the router and that allocates all of the local IP addresses. Simplistically, the switch will look at the IP addresses of connected devices and remember which port they are connected to, then when a packet arrives it will check the destination IP address and if it is one that is sitting on a port, it will send the packet directly there. If it does not recognise an IP address it will ask the DHCP server for information and then route accordingly.
Also, when you want to add another 8 ports, the new switch connects to the original.
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