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Schneider Psy 202SPOWERSIn mathematics special shorthand notation is used to facilitate the presentation ofmaterial and to aid in mathematical calculations and proofsOne such shorthand notationinvolves the case where we want to multiply a number by itself once twice or indeed nnumber of times For example suppose we wanted to determine the product of 4 multiplied byitself six timesThis product which we will call y can be represented asThis notation however becomes extremely clumsy if the number of 4 s is say 326Anotation developed to represent this situation is the basis of our first definitionIn this definition n is a positive integer and it is convenient to restrict a to be a real number241aa aaaaa and aaHencegreater than 0 a0According to this definition a4213 333381 10100 and 9090nOccasionally we have cause to refer to the reciprocal of aThe superscripted integer in the notation discussed above is called an exponent or power of a 8 is therefore 8 and we say that we are taking or raising 24 to the eighthThe exponent in 24powerProperties of ExponentsCertain properties follow logically from these two definitionsSuppose we look at themnproduct a aAccording to definition 11 Similarly1Schneider Psy 202STheir product therefore must equal aaaa where a appears exactly nm times Consequently our first property of exponents is 347Consider for example 22From Prop 11 we know that this product is 2A second property of exponents isAgain this property follows rather readily from a consideration of Definition 11First anthappears exactly n times in the product aIf we now take this product to the m power we will32have a appearing exactly nm timesFor example if we evaluate 3 we have 333 63333Often we will encounter the product of two numbers each taken to the same power nnSuppose we have abIn this expression a and b appear exactly n times in their respectiveproductsNote that the terms in this product can be rearranged so that a and b appear pairwiseie ababab where the product of a and b appears n timesConsequently we havethe next property of exponents33333222 333232323236For example 2The last property of exponents which we will consider is perhaps the most important oneand that is632If we write out each term in this expression we will have Consider the ratio 2From elementary arithmetic we know that we can cancel three 2s in the numerator and3633denominator leaving 2222Hence 222Suppose on the other hand we have 362If we write out every term in this expression we obtain22
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