The equations are quite simple but let me try to show you some worked examples but please realize that I am using fictitious numbers.
For a resistive load:
Power = I² x R ----------- (1)
For a reactive load:
Power = I² x Z x cos θ ---------(2), where Z is the impedance (when it is not purely resistive), θ is the phase angle between the voltage and current phasor. (A phasor is a rotating vector that is used to represent a sinusoidally varying quantity)
Let’s say at rated output the amp applies 40V (rms) across an 8 ohms load, and consider a resistive load first:
I = V/R (Ohm’s Law) = 40/8 = 5A, substitute this into the power formula (1),
P = 5x5x8 = 200W
Across a 4 ohms load,
I = 40/4 = 10A,
P = 10x10x4 = 400W
Across a 2 ohm load,
I = 40/2 = 20A,
P = 20x20x2 = 800W
Across an 1 ohm load,
I = 40/1 = 40A,
P = 40x40x1 = 1600W
So the amp doubles down as long it can deliver the current according to Ohm’s Law, hence the term “high current” amp.
For a “cheap” amp that you are talking about, I would not assume it can double down all the way to 1 ohm so let’s assume the worse case, that is, the amp cannot deliver more than what it can deliver into an 8 ohm load.
Applying the same formula, across an 1 ohm load,
P = 5x5x1 = 25W, in other words, the amp that outputs 200W across an 8 ohm load can only output 25W across an 1 ohm load.
Now, let’s consider a reactive load, as 3 dB says some speakers are highly reactive such that the phase angle between the voltage and current can be as high as 75 degrees. In this case we need to use formula (2) above:
That is Power, P = I² x Z x cos θ, so again assuming the amp cannot output current higher than 5A,
P = 5x5x1xcos(75) = 25x0.258819 = 6.47W
Of course the above example represents one of the possible worst case scenarios. In reality even a cheap pro amp can output higher current into an impedance lower than 8 ohms; and very few speakers will be so highly reactive in the power demanding low frequency range.
In fact many amps (
http://www.crownaudio.com/amp_htm/cdi.htm) will approximately double down to 4 ohms or even 2 ohms, after that it would likely begin to approximate “half” (if there is such a term) down to 1 ohm and below as it reaches its current limit if it is not allowed to clip.