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How do I balance an equation using WolframAlpha?

Posted 7 years ago
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I want to find what combination of exponents a, b, and c will result in the desired outcome of dimensional analysis. An example would be (M^1 L^3 T^-2)^a (M^1 L^0 T^0)^b (M^1 L^2 T^-1)^c = (m^0 L^1 T^0), solving for a b and c... Is there a way to do this on Wolfram? Or any other program/calculator?

The equation you gave is a complicated form of essentially a linear equation. It could be solved in the form you've given it if you gave the additional information that M, L, and T were all incomensurable and that you were looking for a, b, c over the integers. But that would be needlessly difficult.

Instead of the equation you wrote, consider these three equations derived from it by separating the "units":

M^a + M^b + M^c == M^0
(L^3)^a (L^2)^c  == L
(T^-2)^a (T^-1)^c == T^0

So we could solve for {a,b,c} in terms of {M,L,T} but the values of {M,L,T} don't matter. You're really talking about this linear system of equations we can be dervied from those above:

{a + b + c == 0, 
3 a + 2 c == 1, 
-2 a - c == 0}

This linear system of equations is the correct way to think of this problem and how you'd want to enter the problem into any calculator

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