But I'm of two minds about it. I love manipulating expressions by hand. It's a relaxing hobby. But it limits the choice of problems that can be solved, which in turn narrows the range of things that can be taught, and even creates a false sense of what is possible in math. And it doesn't reflect how math is used by most people, i.e., with a computer.
I'd rather incorporate more computers into the math curriculum, and maybe merge math and programming into a single subject.
x^3 = 27 (calculating the volume of a cube)
is an application of the equation of
x^y = z (potentiate a number)
which is an application of the equation
f(x) = y (apply a function)
The solution is simply
x = f-inverse(y)
if f is invertible.
As this simple example shows math on high school level can not splitted up into geometry, algebra, calculus, etc. Understanding one of these areas helps to understand the others and vice versa. If you want to master one of them you have to master all of them at the same time, with the same speed, parallel.
Here's an idea for a better problem:
Solve the following:
x^3 = 27
y^3 = 21
z^4 = 85
My rationale is that there will be a huge time advantage for the student who works out the solutions by using roots, and a visual "hint" that there might be a general solution.