FREE SCRIPTING GUIDE
8.4. Operator Precedence
In a script, operations execute in order of precedence: the higher precedence operations execute before the lower precedence ones. [1]
Table 81. Operator Precedence
Operator  Meaning  Comments 

HIGHEST PRECEDENCE  
var++ var  postincrement, postdecrement  Cstyle operators 
++var var  preincrement, predecrement  
! ~  negation  logical / bitwise, inverts sense of following operator 
**  exponentiation  arithmetic operation 
* / %  multiplication, division, modulo  arithmetic operation 
+   addition, subtraction  arithmetic operation 
<< >>  left, right shift  bitwise 
z n  unary comparison  string is/isnot null 
e f t x, etc.  unary comparison  filetest 
< lt > gt <= le >= ge  compound comparison  string and integer 
nt ot ef  compound comparison  filetest 
== eq != ne  equality / inequality  test operators, string and integer 
&  AND  bitwise 
^  XOR  exclusive OR, bitwise 
  OR  bitwise 
&& a  AND  logical, compound comparison 
 o  OR  logical, compound comparison 
?:  trinary operator  Cstyle 
=  assignment  (do not confuse with equality test) 
*= /= %= += = <<= >>= &=  combination assignment  timesequal, divideequal, modequal, etc. 
,  comma  links a sequence of operations 
LOWEST PRECEDENCE 
In practice, all you really need to remember is the following:
The "My Dear Aunt Sally" mantra (multiply, divide, add, subtract) for the familiar arithmetic operations.
The compound logical operators, &&, , a, and o have low precedence.
The order of evaluation of equalprecedence operators is usually lefttoright.
Now, let's utilize our knowledge of operator precedence to analyze a couple of lines from the /etc/init.d/functions file, as found in the Fedora Core Linux distro.
while [ n "$remaining" a "$retry" gt 0 ]; do # This looks rather daunting at first glance. # Separate the conditions: while [ n "$remaining" a "$retry" gt 0 ]; do # condition 1 ^^ condition 2 # If variable "$remaining" is not zero length #+ AND (a) #+ variable "$retry" is greaterthan zero #+ then #+ the [ expresionwithinconditionbrackets ] returns success (0) #+ and the whileloop executes an iteration. # ============================================================== # Evaluate "condition 1" and "condition 2" ***before*** #+ ANDing them. Why? Because the AND (a) has a lower precedence #+ than the n and gt operators, #+ and therefore gets evaluated *last*. ################################################################# if [ f /etc/sysconfig/i18n a z "${NOLOCALE:}" ] ; then # Again, separate the conditions: if [ f /etc/sysconfig/i18n a z "${NOLOCALE:}" ] ; then # condition 1 ^^ condition 2 # If file "/etc/sysconfig/i18n" exists #+ AND (a) #+ variable $NOLOCALE is zero length #+ then #+ the [ testexpresionwithinconditionbrackets ] returns success (0) #+ and the commands following execute. # # As before, the AND (a) gets evaluated *last* #+ because it has the lowest precedence of the operators within #+ the test brackets. # ============================================================== # Note: # ${NOLOCALE:} is a parameter expansion that seems redundant. # But, if $NOLOCALE has not been declared, it gets set to *null*, #+ in effect declaring it. # This makes a difference in some contexts. 
To avoid confusion or error in a complex sequence of test operators, break up the sequence into bracketed sections.

Notes
[1]  Precedence, in this context, has approximately the same meaning as priority 