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You write e^iπ and only the i ends up in the exponent. If you've ever written LaTeX, you've done this. In Corca you select iπ, right-click, and move it to the exponent.

To differentiate ∛x, you first rewrite it as x^(1/3). Select it, right-click, and pick Change notation. It switches in place, and you carry on from there.

(x² − 1)/(x − 1) looks like it should simplify, and it does. Select the top, right-click, and pick factor. It becomes (x − 1)(x + 1), and you can see the x − 1 that cancels.

Expanding (a + b)³ by hand is where a coefficient usually goes missing. Right-click it, pick expand, and you get a³ + 3a²b + 3ab² + b³.

mx = −kx isn't Newton's law for a spring until the first x has two dots on it. Select it, right-click, and pick double dot. Now it reads mẍ = −kx.

v = v0 + at needs its 0 as a subscript. Select the 0, right-click, and pick to index, and it's v₀. That's the whole menu for now. If there's something else you'd want a right-click to do, tell us.

Not for right click, but have you considered inline maths suggestions just like vscode? I think that could be huge, especially for teachers who have to write common expressions by hand