Exponents base exponent power exponentiation


 Oswald Bridges
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1 Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily 2 by itself 3 times: 2 x 2 x 2 The usul nottion for this oertion is 2 3. We cll 2 the bse nd 3 the exonent (or ower). The oertion itself is clled exonentition. We cn do this with other irs of numbers: 3 = x x 3 5 = 3 x 3 x 3 x 3 x = 2 x 2 x 2 x 2 x 2 x 2 x 2 In fct, though we don t do so here, it cn be shown tht it is ossible to mke sense of exonentition b with ny ir of numbers (including frctions nd so on) nd b, s long s the bse is ositive. The usul nottion for the oertion of exonentition is unlike the nottions we ve seen for the oertions of ddition nd multiliction, where we wrote the symbol for the oertion (+ or x) between the numbers. Insted, for exonentition we ve used suerscrit. This cn be inconvenient when tying into comuter or in other situtions where ll symbols must be on the sme line, so there is nother nottion for exonentition which is commonly used. We write ht symbol, ^, (lso know s cret ) between the numbers to symbolize the oertion of exonentition: ^ b = b As with ddition nd multiliction, there re roerties of the oertion of exonentition. There is no Lw of Commuttivity for this oertion (since 2^3 3^2, tht is, 8 9) but there is Lw of Identity (since ^ = for ny number ). And there re other roerties tht re unique to exonentition. For exmle, if we multily 2 3 x 2 this is the sme s (2 x 2 x 2) x (2 x 2 x 2 x 2) so we multily 2 by itself 3 + times. In other words, 2 3 x 2 = 2 3+ In fct, this is true in generl. For ositive bse nd owers nd q, we hve
2 Rule x q = +q In words, when multilying two numbers with the sme bse, we dd the exonents. Similrly, when dividing two numbers with the sme bse we subtrct the exonents: q Rule 2 q Since, for instnce, we cn cncel 2x2x2 in the numertor nd denomintor below 5 2 2x2x2x2x2 = 2 x 2 = x2x2 Another roerty of exonentition involves rising bse to ower to get certin result, nd then rising tht result to nother ower. For exmle, rise 6 to the ower of 2 to get 36: 6 2 = 36 nd then rise 36 to the ower of : (6 2 ) = 36 To see the generl rule here, note tht (6 2 ) = 6 2 x 6 2 x 6 2 x 6 2 nd by Rule 6 2 x 6 2 x 6 2 x 6 2 = Tht lst exonent is just 2 x, so (6 2 ) = 6 2x In other words, we multily the exonents 2 nd. The generl rule looks like: Rule 3 ( ) q = xq If we rise frction to ower, we cn just the numertor nd denomintor sertely: = x x x x2x2x2 nd multilying the frctions we hve x x x = x3x3x3 or, in generl, 2 3 = Rule b b As mentioned bove, in generl ower cn be ny rel number. This oses the question: wht is some bse to the ower of zero?
3 If we wnt to understnd something like 2 0 we cn t think of this s multilying 2 by itself 0 times. Insted we set 2 0 = It reson it hs to be is to mke our other rules work correctly. If we multily 2 0 x 2 then Rule sys tht 2 0 x 2 = 2 0+ tht is, 2 0 x 2 = 2. Dividing both sides of this lst eqution by 2, we cncel the 2 s to get 2 0 =. In generl, Rule 5 0 = We lso hve to mke sense of negtive exonents. For instnce, wht is 2 ? To mke our other rules work correctly we must set 2  = ½ becuse when we multily 2  x 2 Rule sys 2  x 2 = 2 + tht is 2  x 2 = 2 0 then from Rule x 2 =. Dividing both sides of this lst eqution by 2 (which is 2) we hve 2  = ½ In generl, Rule 6  = Using Rule 6 together with Rules 3 nd, we cn mke sense of ny negtive exonent: so we hve x ( ) Rule 7 = To summrize:
4 Rules of Exonents x q = +q q ( ) q = xq b b q 0 =  = = We will use the Rules of Exonents in the next section to do clcultions with lrge numbers. Now we will see how to use the Rules of Exonents to estimte owers of 2. Powers of 2 come u lot in Informtion Technology, s we will see lter. Here re the first 0 owers of 2: Powers of Becuse owers of 2 come u lot in Informtion Technology, it s worth trying to memorize t lest those of the bove tble. But suose you wnted to know the vlue of 2 3 We certinly don t wnt to memorize the first 3 owers of two! But if we just wnt close ide of how much 2 3 is then we cn use little trick. Notice from the tble bove tht 2 0 is very close to 000 (which is 0 3 ). In symbols we write (the mens roximtely equls ). Now we cn write 2 3 = nd Rule of exonents then sys 2 3 = 2 0 x 2 0 x 2 0 x 2 0 x 2 3 Aroximting ech 2 0 by 0 3 we hve x 0 3 x 0 3 x 0 3 x 2 3 so x3 x 2 3 which mens x 8. Normlly we reverse the order of multiliction nd write
5 2 3 8 x 0 2 The formt 8 x 0 2 is clled scientific nottion nd we will study it in the next section. All of this illustrtes: Method for roximting owers of two To estimte 2 n for some nturl number n, write N = m x 0 + r Where m is the number of times tht 0 goes into N nd r is the reminder. Then 2 n 2 r x 0 mx3 If you memorize the first ten owers of 2 you cn, using this method, quickly estimte ny ower of 2 in your hed.
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