EVOLUTION .] 



MATHEMATICS. ALGEBRA. 



4C5 



denoting a power, is to be divided by the same quantity 

 with a smaller exponent, the subtraction of the latter 

 exponent from the former is all that is necessary ; thus 



^!=a 5 2 = a 3 ; -^=a 3 2 = o' = a, <tc. 

 a" a" - 



Now, this mode of proceeding has suggested tho exten- 

 sion of notation adverted to above, giving rise to sucl 

 expressions as a", a ', a *, <fec. : thus, carrying out 

 the foregoing principle, we have 



- r =ui 1 '=0 : but ~ T or =1, .'. a"=l, a remarkable result. 

 a 1 a 1 a 



a" a" 1 1 



In like manner, -pa" 1 = a * : but = ,- .'. a 1 = 



'a 1 a a 1 a 



Similarly, j- = a * * = a ' 



And in the same way, a * = , o * = ; and 



generally, = a " ; whatever whole number n may 



a" 



stand for. We thus see that without making any inquiry 

 as to the particular values of m and n, we may always 



write 



o ": if m = 3and 



6, then = a , 

 o s 



which, as seen above, is only another way of writing j . 



It may here be observed that unity or 1, divided by any 

 quantity, is called the reciprocal of that quantity ; 



thus, is the reciprocal of o 2 ; is the reciprocal of o ; 



a- a 



and so on. And any quantity with a negative exponent 

 stands for the reciprocal of that quantity with a positive 



exponent ; thus, a * means , a * means - , <tc. ; 



* 



so that not only powers and roots, but also the reciprocals 

 of powers and roots, are represented by exponents. 



In the instances hitherto given of fractional exponents, 

 the numerator of tho fraction has always been unit or 1, 

 the notation implying a root of the quantity to which 

 the exponent is attached. When a power of this root is 

 to be indicated, or the root of a power, tho somewhat 

 cumbersome form used at page 458 namely, (a 1 )* for 

 the third root of the fourth power of a, or (aty for tho 

 fourth power of the third root of a, is not the notation 

 usually employed ; a single fraction is made to serve ; 

 the numerator of the fraction denoting the power, and 

 the denominator tho root ; thus a* would stand in- 

 differently for tho third root of the fourth power of a, 

 or for the fourth power of the third root of a. Whether 

 you regard the power to be taken first and then the root, 

 or the root first and then the power, is of no moment ; 

 the result is the same. Thus suppose you have 8' ; if 

 you regard this as the second power (or square) of the 

 cube or third root of 8, then since the cube root of 8 is 2 

 (seeing that 2 3 = 8), you have 8$ = 2 2 = 4 ; but if, on 

 the contrary, you regard it as the cube root of the square 

 of 8, then since tho square of 8 is 64, you have 8 s = 

 64^ = 4, as before (seeing that 4 3 = G4). Or take the 

 more general case noticed above ; namely, a^. If you 

 regard this as (a 4 )', you consider it the same as (aaaa)^- 

 And if you regard a* as (a*), you consider it tho same 

 as o^aiaiai. Now these two results differ only in ap- 

 pearance ; for let o', that is, the cube root of o, what- 

 ever it be, bo denoted by c ; then, of course, awe 3 ; and 

 it is plain that (eWV)^ = cccc ; tho first side of this 

 ideiit "iv being the former of the above ex- 



pressions, and the second side the latter. It is therefore 



m 



matter of choice whether you call a" the nth root of 



i_ 



*", or tho mth power of a" ; and, in actual numbers, 

 VOL. L 



whichever of these two views is most convenient for the 

 purposes of arithmetic, may be taken. 



Whenever an exponent in tho form of a fraction is 

 such that the numerator and denominator are the same, 



m 



as in a" 1 , that exponent may be replaced by unit, or 1 ; 

 because if the mth power of any quantity be taken, and 

 then the mth root of the result, the original quantity (in 

 this case a) is of course brought back again. 



The second operation merely undoes what the first 

 does ; tho two operations mutually destroy one another, 



n 



and are .'. of no effect, so that o=a. In consequence 



of this, such an expression as a' "is the same as o'' 

 since the former means the mth root of the mth power 

 of this. When a fractional exponent, therefore, has a 

 factor common to both numerator and denominator, tho 

 common factor may be expunged. All that is said above 

 applies as well to negative as to positive exponents, as 

 each expression there considered may be equally regarded 

 as the denominator of a fraction whose numerator is 1. 

 The following is a view of the principal operations with 

 exponents when they are attached to the same quantity : 

 pm p 



a va =a q ; for example, a S =a *=a ? ; a*=a".l=a 9 ; &c. 

 a m -X.a n =a m +"; for example, aXa 9 =a 5 ; a ? 'Xa^= ?1 -(- i =''' i 

 i m -i-a"=a m ~"; for example, a s -a-=a :l ; a^a^=a^^=a^ 



a m y > =a mn ; for example, (a 3 ) 2 =a; (aty=a* j &c. 

 That is 



To MULTIPLY. Add tlie exponents. 



To DIVIDE. Subtract ttc exp. of divisor from exp. of 

 dividend. 



To EXPRESS A POWEB on ROOT. Multiply the ei-p. 

 of the quantity, by the exp. of the power or root of it which 



l'i In-',. ,, . /. 



In evolution, or the extraction of roots, there are some 

 particulars respecting signs which require to bo especially 

 mentioned. In involution, or tho raising of powers, 

 you have seen that the sign of tho result is always fixed 

 by tho rule of signs ; it is not so in the reverse operation 

 of extraction. For instance, the square of 4 is 16, tho 

 16 being plus, whether the 4 be plus or minus; but tho 

 square root of 16 is ambiguous as to sign ; the numerical 

 value of the root is, of course, 4 ; but wo have as much 

 right to prefix a minus to this 4 as a plus ; since ( 4) a 

 and 4 2 are equally 16. Hence *J 16 = + 4 ; that is 

 ylus or minus 4. And there is a like ambiguity, as to 

 iign, in every even root of a positive quantity ; because 

 the corresponding even power of that root would be tho 

 same whether a + or a be prefixed to it : ,J 4 or 4J- ia 

 -f 2 ; \/ 16 or 16 j is + 2, and so on. As to an even root 

 of a negative quantity, the thing is impossible. Such an 

 expression as J 4 implies an impossible operation ; 

 'or the srjuare root of a quantity is that which, when 

 squared, reproduces the quantity. Now a quantity 

 squared, whether its sign be + or , is always 4- ; it 

 a impossible . '. that 4 can be tho smiare of anything, 

 such expressions as J 4, J 9, J 1, <tc., arc /. 

 called imaginary or impossible quantities ; and they an- 

 swer this purpose, namely whenever they occur in tho 

 solution of a problem, you may take it as a sure indica- 

 ,ion that tho problem implies some impossibility or con- 

 ;radiction. You thus see that the numbers of arithmetic, 

 when introduced into algebra, divide themselves not only 

 nto positive and negative, but also into real and imaginary. 

 teal numbers, too, are separated into two classes ; 

 namely, rational and irrational, or surd. Tho following 

 are examples of surds; namely, / 2, ^3, / 5, ,J 7, 

 ^/ 10, <kc. Surds, you see, are rooty; but roots of 

 numbers that are not themselves the reverse powers : 3, 

 , 7, 10, <fee., are not squares; they have .'. no exact 

 qiiare roots. Nevertheless, either of those numbers 

 >eing proposed, wo can always find another number, 

 luch that its square shall approach as near to the pro- 



3o 



