Eli5: Why are exponents so common in scientific formulas? Why is it so common that numbers in nature are multiplied by themselves?
In: Mathematics
The answer lies in how the numbers are used. Take, for instance, the area for a volume of something. For a cube, you multiply length times width times height. This is intuitive. Doing so gets you something like 2m long, 4m wide, 3m tall for a total of 24m^3.
This is NOT one measurement multiplied by itself three times, but three separate measurements all taken according to the same standard (a meter stick) and then combined (multiplied) to tell you the volume of the object.
The volume of a sphere is less straightforward, but results in (4/3)*pi*r^3. Why is the radius cubed here? Because the length, width, and height of the sphere all relate directly to the radius. In this special case, the three measurements are the same, but similar formulas that calculate the volume of an egg require three unique measurements to be taken of a spherical(ish) egg.
For a more advanced example, consider Power. Well, power is measured in watts. Watts are defined as being energy over time, or rather, the force required to move an object multiplied by the velocity of the object – if a device requires 10W to run, that is 10 Newtons of force constantly being applied to a 1kg object traveling at 1m/s.
Where do the exponents come from? Because Newtons themselves (force) is equal to the mass of an object times acceleration…
N=kg*m/s^2
So power can be rewritten as
P=F*v
P=(kg*m)/(s^2)*(m/s)
P=(kg*m^2)/s^3
Why so many exponents?
Because in order to measure the power being applied to the object, you need to know one mass, two lengths, and three separate measurements of time.
Sometimes, some measurements happen to be the same (like in the volume of a sphere). Other times, you need 6 different measurements to get power.
All other formulas are similar. Each measurement adds one variable. Similar variables are combined. That doesn’t mean the measurements are being multiplied by themself, only that the measurements are measuring similar things (lengths, times, masses, etc).
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