algorithm analysis because the running times of loops naturally give rise to summations using summationwe can rewrite the formula of proposition as likewisewe can write polynomial (nof degree with coefficients as thusthe summation notation gives us shorthand way of expressing sums of increasing terms that have regular structure the exponential function another function used in the analysis of algorithms is the exponential functionf(nbnwhere is positive constantcalled the baseand the argument is the exponent that isfunction (nassigns to the input argument the value obtained by multiplying the base by itself times in algorithm analysisthe most common base for the exponential function is for instanceif we have loop that starts by performing one operation and then doubles the number of operations performed with each iterationthen the number of operations performed in the nth iteration is in additionan integer word containing bits can represent all the nonnegative integers less than thusthe exponential function with base is quite common the exponential function will also be referred to as exponent function we sometimes have other exponents besides nhoweverhenceit is useful for us to know few handy rules for working with exponents in particularthe following exponent rules are quite helpful proposition (exponent rules)given positive integers , ,and ,we have (ba) bac babc ba+ ba/bc ba