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The problem in the posed question is the "mass" in the equation you quote is the mass of the object upon which the force (whether it be a photon or not) is acting, NOT the mass of the object exerting the force.

You can MEASURE the net force on an object with mass simply by measuring the acceleration of that object and dividing it by the object's mass.

Or you can predict an acceleration of an object with mass by calculating what

its net force will be, and then dividing that by the object's mass.

Unrelated to the above excellent answer, but another comment on the question:

You mention, correctly, that photons have no rest-mass. But the photon is never

at rest, and at the speed at which it moves from place to place, it has mass.

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Q: How do photons exert force if force is mass times acceleration and photons have no rest mass and travel at constant speed?
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