Ihave been trying to understand why the diffie hellman key exchange algorithm works, specifically why the two exponents can be swapped in it without the result changing.

so my specific question is why:

(g^b mod m)^a mod m
= (g^a mod m)^b mod m

i know it is "because" it can be correctly rewritten as the trivially true equation:

g^(ab) mod m
= g^(ba) mod m

but that doesn't explain why it is true, i am interested in the reasoning/proof/explanation behind why the first of these equivalencies hold true.

you

For every corresponding pair of cross sections, the area of the cross section of a sphere with radius r is equal to the area of the cross section of a cylinder with radius and height 2r minus the volume of two cones, each with a radius and height of r. a cross section of the sphere is and a cross section of the cylinder minus the cones, taken parallel to the base of cylinder, is the volume of the cylinder with radius r and height 2r is and the volume of each cone with radius r and height r is 1/3 pie r^3. so the volume of the cylinder minus the two cones is therefore, the volume of the cylinder is 4/3pie r^3 by cavalieri's principle. (fill in options are: r/2- r- 2r- an annulus- a circle -1/3pier^3- 2/3pier^3- 4/3pier^3- 5/3pier^3- 2pier^3- 4pier^3)