I need to "tap" into an existing split circuit receptacle to run power to my new dishwasher. Can somebody provide some advice please?

Buster

 

I've got the same thing under my sink. The dishwasher plugs into the uncontrolled side, and the switched side runs the disposall.

Do you have a specific question?

 

Yah. Find out all that's on the load to begin with. That shouldn't be hard. Wire to the unswitched side. Make sure that it's not two circuits sharing a neutral. If it is make sure both are on different phases, or better yet avoid it completely.

 

After turning the breaker off, check outlet with a known good meter to make certain the outlet is dead. THEN take it apart, not before.

The correct tools for your specific application (Romex tool, or MC splitter, or whatever fits your details) may seem expensive - but the frustration they will save make them worth every penny.