Starting with your list
list = Table[RandomChoice[{a, b, c, d}, 2], {4}]
(* {{b, c}, {c, d}, {b, b}, {c, d}} *)
I often find it useful to break it down in steps.
Step 1. Compute the cube
Map[#^3 &, list]
(* {{b^3, c^3}, {c^3, d^3}, {b^3, b^3}, {c^3, d^3}} *)
Step 2. Apply Plus
Here is a brute force way to use Map
on the previous output. @@
is a shortcut for Apply
.
Map[Plus @@ # &, Map[#^3 &, list]]
(* {b^3 + c^3, c^3 + d^3, 2 b^3, c^3 + d^3} *)
However there is a nice shortcut shown in kglr's comment. @@@
is a shortcut for Apply
acting upon heads located at level 1.
Plus @@@ Map[#^3 &, list]
(* {b^3 + c^3, c^3 + d^3, 2 b^3, c^3 + d^3} *)
Step 3. Compute the square root
By now I think you get the idea. We will use Map
to compute the square root of the outputs of step 2.
Map[Sqrt[#] &, Plus @@@ Map[#^3 &, list]]
(* {Sqrt[b^3 + c^3], Sqrt[c^3 + d^3], Sqrt[2] Sqrt[b^3], Sqrt[c^3 + d^3]} *)
In the notebook one sees

Sqrt[Plus @@@ (#^3)] &[list]
orSqrt[Total[Transpose[#^3]]] &[list]
? $\endgroup$☺ = +## & @@@ (#^3)^(1/2) &; ☺ @ list
:) $\endgroup$