Suppress trailing 0s in numerical values

Other answers have provided solutions but they have not directly addressed why this happens. The use of N on exact values with a numeric second parameter (e.g. 10) produces arbitrary-precision output, and these values are by default displayed in full without truncation.

Compare:

5.000000   (* machine precision Real *)
5`7        (* arbitrary precision *)

5.

5.000000

The methods in Alan's and Michael's answers get around this by converting to machine precision and then adjusting output. However if you want to print more than $MachinePrecision digits these will not work. Carl's answer comes closer but it also has a problem in that more digits than requested may be printed.

If you specifically want to suppress trailing zeros then you could process to affect that.

For whole numbers simply:

rep = r_Real /; r == Round[r] :> N @ Round[r];

N[Table[4^(i/4), {i, 1, 4}], 25] /. rep

{
 1.414213562373095048801689,
 2.,
 2.828427124746190097603377,
 4.
}

If you need to preserve the arbitrary precision nature of the output when used as input use Interpretation:

rep2 = r_Real /; r == Round[r] :> Interpretation[N @ Round[r], r];

If you want to trim trailing zeros that are not immediately following the decimal point you might turn to string processing:

rep3 = r_Real :> 
   Interpretation[
    RawBoxes @ StringReplace[ToString[r], 
      Shortest[x__] ~~ "0" .. ~~ EndOfString :> x], r];

N[Table[4^(i/4), {i, 1, 4}]/10, 25] /. rep3 // Column

0.1414213562373095048801689
0.2
0.2828427124746190097603377
0.4

If you want to do this globally for a session consider $PrePrint, e.g.

$PrePrint = # /. rep3 &;

Then you can simply write:

N[Table[4^(i/4), {i, 1, 4}]/10, 10]

NumberForm[N@Table[4^(i/4), {i, 1, 4}], 10]

You can create a form that displays numbers the way you want:

MakeBoxes[SuppressedZeroForm[a_], StandardForm] ^:= MakeBoxes[
    Defer[InputForm[a, NumberMarks -> False]]
]

For example:

N[Table[4^(i/4), {i, 1, 4}], 10] //SuppressedZeroForm

{1.4142135624, 2., 2.8284271247, 4.}