Using sets of arguments from a list

You could use Apply at the first level:

Times @@@ listA

Short answer: See Heike's post.

Longer version:

There are a couple of functions in the functional programming toolbox that are so common that they've gotten their own symbols. One of them is Apply, which takes a list and uses its elements as function arguments:

Apply[f, {a, b, c}]

f[a, b, c]

The shorthand notation for this would be f @@ {a, b, c}.

Additionally, Apply has an optinal third argument, specifying the depth at which the function should be applied:

(* Apply on 0-th level *)
Apply[f, {{a,b}, {c,d}}]

f[{a,b}, {c,d}]

(* Apply on the first level *)
Apply[f, {{a,b}, {c,d}}, {1}]

{f[a,b], f[c,d]}

The latter is also a very common expression, it has the shorthand notation f @@@ {{a,b}, {c,d}}. The notation ends here, i.e. there's no @@@@; if you need that, you'll have to use Apply explicitly. The documentation features plenty of examples if you need further help.

That said, what you want is converting a list of lists of numbers to a list of the product of the numbers. This is equivalent to applying the product function on the first level of the original list; that function is called Times in Mathematica (Product stands for the mathematical expression $\prod_a^bx$). Times is the internal function that is called when you enter something like a*b*c: it becomes Times[a,b,c] internally. Therefore, using the Apply function from above,

Apply[Times, {{a,b}, {c,d}}, {1}]

(* Evaluates to ... *)
{Times[a,b], Times[c,d]}

(* ... and is equivalent to ... *)
{a b, c d}

That's precisely what you want. Adding the syntactic sugar from above, @@@, this leaves you with the final short notation

Times @@@ list

to solve the problem.

Just for fun:

Replace[list, {x__} :> 1 x, 1]

Since this answer has garnered some votes here is a more direct form of the implicit Times:

1##& @@@ list