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Smaller gates mean lower capacitance, so you need to drive less current into them to switch.

The nice thing about CMOS is that you only need to flow current when switching them (at other times current is blocked by an "OFF" transistor somewhere in the logic), so resistance doesn't matter as much as it would for a motor or a light.

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This is not entirely true, in an ideal world a CMOS device just sitting there not switching would use basically no power, but in actual application there is some leakage (both form not being fully off and some current jumping from the gate). This leakage gets worse as the transistors get smaller and when you have billions of transistors even the tiniest amount of leakage adds up, the result is leakage current can be a significant amount of the power used by a modern processor.

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