What is the overall effect of a change in resistance on the power?
According to P = V^2/R, increasing the resistance decreases the power. This intuitively makes sense. If we hold voltage constant and increase the resistance, less current is supplied to the load and power decreases.
But, according to P = I^2R, increases the resistance increases power. Funny thing that this also can be rationalized. We hold current constant and increase the resistance. This increase the voltage, and thus power.
W/o knowing what to hold constant, can we identify whether power increases or decreases?
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In short, no. Any real system is limited either by voltage or current. Because current and voltage are also related by P = IV, you basically have to know which one is restricted in order to determine how the power output will change. This is logical: all four values are intricately linked.
Usually, you generate current by applying a voltage difference, so you can assume that the constant value in most real systems is voltage (this is true both of electricity from the plug and from a battery).
When you increase resistance while keeping voltage constant then Ohm's law says that current will decrease but when the same is done while keeping current constant then voltage will increase which in turn increases power.