60 



THE POPULAR EDUCATOR. 



snuff-boxes. 28. We feave a press, seven beds, and nine looking- 

 glasses. 29. There are twelve months in a year, seven days in a week. 

 30. There are four weeks and two or three days in a month. 31. In 

 our school there are ten forms. 32. Three times four are twelve. 

 33. Three times three are nine. 



EXEECISE 27. 



1. Gli amici di mio zio sono ricchissimi. 2. Ho spesso veduto 

 questi uomini. 3. I fanciulli della nostra giardinera sono ragionevoli. 

 4. Abbiamo trovato le sorelle d' Enrico nella chiesa. 5. Questa 

 madre e sempre contenta, ma le nostre vicine sono spesso malcontente. 

 6. I vostri temi sono difficili ma i temi di Luigi sono molto 

 facili. 7. Avete voi ricevuto questi bei fiori da Giovanni ? 8. II 

 nostro cugino ha tre tabacchiere. 9. Ho ricevuto da mio zio un 

 tomperino e venti penne. 10. L' arnica di mia sorella ha cinque cufie. 

 11. Questa signora ha sette fanciulli. 12. Ho comprato due specchj e 

 sei sedie. 13. Questo uoino ha quattro flgli e due flglie, che sono 

 molto ragionevoli. 14. Abbiamo ricevuto cinque lettere da nostra zia. 

 15. II mio amico ha trovato un temperino e otto penne. 16. Ho 

 perduto neHa scuola dieci penne. 17. Cinque via quattro venti. 



LESSONS IN ALGEBRA. XXVII. 



DIVISION OF POWERS. 



POWERS may be divided, like other quantities, by rejecting 

 from the dividend a factor equal to the divisor ; or by placing 

 the divisor under the dividend, in the form of a fraction. Thus 

 the quotient of o, 3 6- divided by 6 2 is a 3 . 



EXAMPLE. 'The quotient of a 5 divided by a* is . But 

 this is equal to a 2 . For in the series 



a 4 , a 3 , a 2 , a 1 , a, a" 1 , a~", a~ 3 , a~ 4 , etc., 



if any term be divided by another, the index of tho quotient 

 will be equal to the difference between tho index of the dividend 

 and that of tho divisor. 



aaaaa, a m 



Thus a, 5 -f- a 8 = = a 2 ; and a m -r a u = --? = a m - n . 

 aaa a 11 



Hence we deduce the following 



GENERAL RULE FOR DIVIDING POWERS. 



A power may be divided by another power of tlie same root by 

 subtracting the index of the divisor from that of the dividend. 



If the divisor and dividend have co-efficients, the co-efficient of 

 the dividend must be divided by that of the divisor. 



If the divisor and dividend are both compound quantities, the 

 terms must be arranged, and the operation conducted in the same 

 manner as in simple division of compound quantities. 



EXAMPLE. Thus y* -r y 2 = i/ 3 ~ a = i/ 1 . That is, = y. 

 [The above rule is equally applicable to reciprocal powers.] 



EXERCISE 45. 



6. Divide x" by x". 



7. Divide y m by jy. 



8. Divide b 6 by b 3 . 



9. Divide 8a + by 4a. 



10. Divide a + 3 by a 2 . 



11. Divide 12(b + y) D by 3(b 4- y) s . 



1. Divide 9aV by - 3aS - 



2. Divide Wx" by Zb 3 . 



3. Divide a-b + 3a 2 y* by a 2 . 



4. Divide d x (a-h + y) 3 by (a - 



fe+i/)3. 



5. Divide a+ 8 by a. 



BOOTS. 



If we resolve b s , or bbb, into equal factors, viz., b, b, and b, 

 each of these equal factors is said to be a root of & J . So if we 

 resolve 27 into its three equal factors, as 3 X 3 X 3, each of 

 these equal factors is said to be a root of 27. And when any 

 quantity is resolved into any number of equal factors, each of 

 those factors is said to be a root of that quantity. 



A root of a quantity, then, is a factor which, multiplied into 

 itself a certain number of times, will produce that quantity. 



The number of times the root must be taken as a factor to 

 produce the given quantity, is denoted by the name of the root. 



Thus 2 is the fourth root of 16 ; because 2x2X2x2 = 1 6, 

 where 2 is taken four times as a factor to produce 16. 



So a 3 is the square root of a 6 ; for a 3 X a. 3 = a 6 . 



Powers and roots are correlative terms. If one quantity is a 

 power of another, the latter is a root of the former. As b 3 is the 

 cube of b, so b is the cube root of b 3 . 



There are two methods in use for expressing the roots of 

 quantities ; one by means of the radical sign V > & n d the other 

 by a fractional index. The latter is generally to be preferred ; 

 but the former has its uses on particular occasions. 



When a root is expressed by the radical sign, the sign is 

 placed before the given quantity, in this manner, V a - 



Thus 2 V& is the 2nd, or square root of a ; 3 ^fa is the 3rd, or 

 cube root. 



The figure placed over the radical sign denotes the number of 

 factors into which the given quantity is resolved ; i.e., the 

 number of times the root must be taken as a factor to produce 

 the given quantity. 



Thus 2 v f O' 2 shows that a 2 is to be resolved into two factors, 

 and 3 ^* 3 into three factors, and n *Ja n into n factors. 



The figure for the square root is commonly omitted, and the 

 radical sign is simply written before the quantity. Thus V <* 2 



When a figure or letter is prefixed to the radical sign without 

 any character between them, the two quantities are to be con- 

 sidered as multiplied together. 



Thus 2\fa is 2 X V a 5 that is, 2 multiplied into the root of 

 a ; or, which is the same thing, twice the root of a. 



And x>Jb is x X V&> or x times the root of b. 



When no co-efficient is prefixed to the radical sign, 1 is 

 always understood; \/ a being the same as 1V&; that is, once 

 the root of a. 



The cube root of a, 6 is a 2 ; for a 2 X a 2 X a 2 = a 6 . 



Here the index is divided into three equal parts, and tho 

 quantity itself resolved into three equal factors. 



The square root of a 1 is a 1 or a ; for a X a = a 2 . 



By extending the same plan of notation, fractional indices 

 are obtained. 



Thus, in taking the square root of a 1 or a, the index 1 is 



i 

 divided into two equal parts, i and \ ; and the root is a?. 



On the same principle, the cube root of a is a 3 = 3 ^fa. 



Tho nth root, is a" = n </a, etc. 



Every root, as well as every power of 1, is 1 ; for a root is a 

 factor, which, multiplied into itself, will produce the given 

 quantity. But no factor except 1 can produce 1, by being 

 multiplied into itself. 



So that 1", 1, Vl> "V*> e t-> are all equal. 



Negative indices are used in the notation of roots, as well as 

 of powers. 



JL = a -i. -l = a -3. -l = a --Ej J__ a -5fi 

 a 2 a 3 a 'n a'ian 



POWERS OF ROOTS. 



In the preceding examples of roots, the numerator of the 

 fractional index has been a unit. There is another class of 

 quantities, the numerators of whoso indices are greater than 1 ; 



as fc 3 , c*, etc. These quantities may be considered either as 

 powers of roots, or roots of powers. 



N.B. In all instances, when the root of a quantity is denoted 

 by a fractional index, the denominator, like the figure over the 

 radical sign, expresses the root, and the numerator the power. 



Thus a? denotes the cube root of the first power of a ; i.e., that 



i i i 

 a is to be resolved into three equal factors ; for a 3 X a 3 X a' 



3 



= a. On tho other hand, c 5 denotes the third power of the 

 fourth root of c, or the fourth root of the third power. One 

 expression is equivalent to the other. 



The value of a quantity is not altered by applying to it a 

 fractional index whose numerator and denominator are equal. 



as? 



Thus, a = a 2 = a 3 = a". For the denominator shows that a is 

 resolved into a certain number of factors ; and the numerator 



n 



shows that all these factors are multiplied together in a". On 

 the other hand, when the numerator of a fractional index 

 becomes equal to the denominator, the expression may bo 

 rendered more simple by rejecting the index, 

 n 



Instead of a", wo may write a. 



The index of a power or root may be exchanged for any other 



index of the same value. 



2 4 



Instead of a?, we may put a 8 . 



For in the latter of these expressions, a is supposed to be 

 resolved into twice as many factors as in the former ; and the 



