You wrote: "Exercise 5 : Implement a Mathematica module, whose input is a string x and an integer n, to compute x^n" but your test was puissance[2,3]

So, I'm not exactly sure what the expected behavior is. However, I can give some feedback that might be helpful.

• In puissance[x, n] := ..., x and n are symbols, not patterns, so you've effectively defined a function for exactly that one case.
• Return res, is not how you apply functions to arguments in Mathematica. That expression will resolve to a multiplication of the symbol Return and the variable res.
• There is really no need to use Return at all, as the result of Module will simply be the last expression in the body.
• string[[i]] is not how you access characters in a string. An error would be produced, and your variable s won't be updated.

Since this is homework, I don't feel comfortable giving you corrected versions. I would suggest you find a tutorial on basic Mathematica syntax. There are many resources at the Wolfram website and in the help documentation from within Mathematica.

Your examples suggest that you're comfortable with an imperative style of code. One thing these exercises might be trying to teach you is to use a more functional style. Mathematica code can be written in many styles, but for these exercises there are very simple, small, functional solutions. I suggest you try to solve the problems without using Module, For, Return, or AppendTo. I know the exercises mentioned "Mathematica module", but I'm not sure if you were quoting exactly anyway, and I'm assuming that "module" was just being used to mean a "solution in code".

If that's too vague, maybe you can provide a bit more context and I can try to be more specific without just giving you the answers.

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We use special formatting for code that you should follow - see HERE how.

I think you need to approach this differently. But I fixed your code so at least it works as I think you intended it to:

puissance[x_, n_] := Module[{res, i},
  res = x;
  For[i = 1, i < n, i++, res = res*x];
  Return[res]]

puissance[3, 3]
(*27*)

characters[string_] := Module[{s = {}, i},
  For[i = 1, i <= StringLength[string], i++, AppendTo[s, StringTake[string, {i}]]];
  Return[s]]

characters["A string"]
(*{"A"," ","s","t","r","i","n","g"}*)

Now you have to understand that Wolfram Language or WL (Mathematica is programing environment) is a functional programming language. You seem to use a lot of procedural (loops, cycles) constructs, are you sure you need to? Here is my solutions to you problems if I understood them correctly.

power[s_String, n_Integer] := ToExpression[s]^n;
power["3", 3]
(*27*)

StringTake["123456789", {3, 5}]
(*"345"*)

This maybe not what you need, but at least it shows how simple is functional form comparing to procedural, because it is native for WL.