MULTIPLICATION.] 



MATHEMATICS. ALGEBRA. 



450 



In the case of a power of a quantity, the exponent is 

 a whole number, which indicates what poiver is meant ; 

 in the case of a root, the exponent is a fraction, the 

 denominator of which indicates wliat root is meant. 

 The power of a proposed quantity may be easily deter- 

 mined, as the multiplication may be actually executed 

 with but little trouble ; but to find a specified root of a 

 proposed quantity even of a common number is often 

 a difficult matter ; some numbers, indeed, have no exact 

 roots. But we shall have to speak about roots again. 



MULTIPLICATION. 



CASE I. When the factors are simple quantities. 



RULE 1. Observe whether the signs of the two factors 

 are like or unlike ; if they are like, that is both + or 

 both , write -f- for the sign of the product ; but if they 

 are unlike, that is, one + and the other , write for 

 the sign of the product. 



2. After the sign, write the product of the coefficients. 



3. After the product of the coefficients, write that of 

 the letters: that is, put down the letters in both factors, 

 one after the other, without any sign between them, 

 and the complete product will be exhibited. 



Thus, if we have to multiply together the two factors, 

 4ox and Sby, we first observe the signs ; these being 

 'If, we know that the sign of the product is minus ; 

 this minus we write 12, the product of the co- 

 efficients, and finally we place against the 12 the quan- 

 tity abjcy, this being the product of the letters when 

 arranged in alphabetical order, /. the product is 

 12abxy. Again, if wp have to multiply Ibxz by 

 5acy, then tin signs being like, we write + for the sign 

 of the product, 35 for the product of the coefficients, 

 and abcxyz for the product of the letters ; .'. 7k<" X 

 5acy = Soabcxyz, the plus sign being omitted, as un- 

 necessary. In a similar way we have 



3mr X 6y = 1 &ouy, 24ey X 4aj = 8a4c.ry. 

 ,-4 a .rXSa : '4.r= 20 a 5 ** 3 , 34z*X 8<i'4.rz=24a'4 3 xr'. 



Y<m will observe that the third of these examples if 

 the same as 4oox X baaabx = ZOaaaaabxx ; and 

 that this result, in the more brief notation for powers, 

 is 20a 6 bx 2 . And you must perceive that, by always 

 adopting this notation, the multiplication of pouxn of 

 the same q\tantity is reduced simply to the addition of 

 the exponents of the factors ; thus, the factors in x 2 x a x 6 , 

 factors which are all powers of the same quantity x, are 

 ', x 6 ; the sum of the exponents is 2-(- 3 + 5 = 30: 

 iiat x^x'x* = x' ; nothing, therefore, can be more 

 easy than the multiplication of powers of the same quitn 

 tity. We need scarcely tell you that x is the same as 

 x 1 , so that xx = x 3 , x 2 x = x 3 , <fcc. 



The preceding direction, as to the sign of the product 

 of two factors, is called the Ride for the signs. It is 

 briefly expressed thus : Like signs give PLUS ; unlike 

 signs, MINI'S. 



You may satisfy yourself that the rule for tlie signs is 

 true as follows : 



Take any two factors whatever, say 7 and 3 ; then all 

 the possible varieties, as to signs, will be these namely, 

 7x3, 7X3, 7X 3, 7x 3. 



The first is the case of common arithmetic, the product 

 being 21 ; the second case requires no consideration ; 

 for 7, repeated three times, is necessarily 21. The 

 third case is peculiar ; but we may arrive at the true 

 product thus : increase the multiplier 3 by 4 ; the 

 product, whatever it bo, will obviously be 4 times 7 

 that is, 28 too great ; but the multiplier increased by 4 

 becomes 1 ; and once 7 is 7, and as this is 28 too great, the 

 correct product must be 7 28, that is 21, .'. 7 X 

 3= 21. 



In like manner, in the fourth case, increase the multi- 

 plier 3 by 4 ; then, as before, the product will be four 

 times 7, that is, 28 too great ; in other words, 

 28 must be subtracted from the erroneous product to 

 make it correct ; but the multiplier increased by 4, is 1, 

 and once 7 is 7 ; .'. the correct product is 7+28, 

 that is 21 ; because the sign of the 28 must be changed 

 when subtracted ; .'. 7X 3=21. You thus see that 



when the factors have like signs, the product is plus ; 

 and that when they have unlike signs, the product is 

 minus ; and, from the foregoing reasoning, it is plain 

 that the same conclusion would have followed if any 

 other two factors had been chosen ; .'. the rule for the 

 signs in multiplication is general. 



Ex. 1. lax 2. 7ax a 



36 36x 



. 



1 2. G& 2 cx" x 36 4 c 2 x = 



13. a 2 2 



3. 



7. 



EXAMPLES FOR EXERCISE. 



[NOTE. You will observe that the rule for the signs 

 enables us to fix the sign of the product of two factors 

 only ; but it is unnecessary that it should do more than 

 this ; if there are three factors, the product of two be- 

 comes a factor to be combined with the third. It is 

 plain that, however numerous the factors when the 

 number of them preceded by the minus sign is odd, the 

 sign of the product is minns ; and that when the number 

 of minus factors is even, the sign of the product is plus.] 



1. 9aVx4a 2 ;/ 3 2. 76xVx86V 



4a 2 x 3 z*X 6 4. 5WyX 3ei/% 



lla**yVx 9aV/s' C. 13a 2 j; V X HaW 

 J','., Vx-4& 2 J; yS 8. iaxV 



9. 2ax 2 i/X 3a 2 i/ 3 X 4a 3 .e 4 



10. 36 2 i/zx 2i/ 2 z 2 x 4a& 3 y 



11. Jax^X&e'yX 3a 2 i/ 2 



12. Vx I-xVX i^** 



13. Ja 2 i/ 2 X 36Vx T<i6x 2 ! 



14. Ja'xx z 2 !/X 2;/ 2 zX 



CASE II. When the multiplicand is a compound quan- 

 tity, and the multiplier a simple quantity. 



RULE. Multiply each simple term in the multiplicand 

 by the multiplier, beginning always at the left hand ; 

 connect the several products together by their proper 

 signs, and the complete product will be exliibited. 



Zaxy 



2. 3<try 2 44r- 



6.r c i/- 2 

 3**' 



2 34"y 



4. Sa-r^y 4y 2 z-)-^a4 



i z 3a : '4V 



5. 



2 \ab-xy- a'bxz* 



