proposition (exponent rules)given positive integers aband cwe have (ba ) bac ba bc ba+ ba /bc ba- for examplewe have the following ( ) * (exponent rule + (exponent rule / / - (exponent rule we can extend the exponential function to exponents that are fractions or real numbers and to negative exponentsas follows given positive integer kwe define / to be kth root of bthat isthe number such that rk for example / since likewise / and / this approach allows us to define any power whose exponent can be expressed as fractionfor ba/ (ba ) / by exponent rule for example / ( ) / / thusba/ is really just the cth root of the integral exponent ba we can further extend the exponential function to define bx for any real number xby computing series of numbers of the form ba/ for fractions / that get progressively closer and closer to any real number can be approximated arbitrarily closely by fraction /chencewe can use the fraction / as the exponent of to get arbitrarily close to bx for examplethe number is well defined finallygiven negative exponent dwe define bd / - which corresponds to applying exponent rule with and - for example - / / geometric sums suppose we have loop for which each iteration takes multiplicative factor longer than the previous one this loop can be analyzed using the following proposition proposition for any integer > and any real number such that and  consider the summation ai an = (remembering that if this summation is equal to an+ - summations as shown in proposition are called geometric summationsbecause each term is geometrically larger than the previous one if for exampleeveryone working in computing should know that - for this is the largest integer that can be represented in binary notation using bits