I only find information on the Kärcher K7 that it has 2'200W. Where is the information about 3'000W from?
Cables are never specified in terms of watts (W). Cables are not devices that provide power. They are specified according to max current (A) and voltage (230V/400).
Cables have no direct influence on a device running at lower power than it could. If it is too small (wire cross-section) it will get warm, hot and could start to burn. The fuse should not blow first.
Where and how does the assumption come about that the cable "brakes" the Kärcher?
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