Mdtryhht

MWE

documentclass[a4paper,11pt,border=1pt]standalone
%LetLtxMacrooldcdotcdot
%renewcommandcdot!cdot! <-- Like this
begindocument
$3cdot 10^8$ m/s
enddocument

I'm getting the following output.

However cdot as there are many spaces between left and right. How do I get the following image should I have a macro?

Related to: https://tex.stackexchange.com/a/29838/33075

For this I recommend siunitx, that ensures uniformity in numbers and units.

documentclassarticle
usepackageamsmath
usepackagesiunitx

sisetup
 exponent-product=cdot, % double brace for avoiding the space
 per-mode=symbol,


begindocument

SI3e8meterpersecond % long form

SI3e8m/s % abbreviated form

SI[per-mode=reciprocal]3e8meterpersecond % long form

enddocument

Note that it's easy to change from a representation to another, when the long form is used (which is recommended).

Also this has the advantage that you just need to change the option in the preamble if you change your mind about how to represent that product.

The same effect can be obtained with the option tight-spacing, which however will act also on all binary operations, for instance uncertainties.

documentclassarticle
usepackageamsmath
usepackagesiunitx

sisetup
 exponent-product=cdot,
 tight-spacing,
 per-mode=symbol,


begindocument

SI3e8meterpersecond % long form

SI3e8m/s % abbreviated form

SI[per-mode=reciprocal]3e8meterpersecond % long form

enddocument

cdot is defined in fontmath.ltx with:

DeclareMathSymbolcdotmathbinsymbols"01

which means it is a binary operator (mathbin) so it will have an extra space before and after as other binary operators, such as + and -.

TeX will not insert that space if you “hide” cdot within braces:

documentclass[11pt,border=1pt]standalone
begindocument
$3cdot 10^8$ m/s
enddocument

If you will use that symbol multiple times you can define an ordinary math symbol (mathord) with the same glyph as cdot:

documentclass[11pt,border=1pt]standalone
DeclareMathSymbolcodotmathordsymbols"01
begindocument
$3codot 10^8$ m/s
enddocument

or you can redefine cdot with the same command.