THE POPULAR EDUCATOR. 



Eighteen thousand Eussians were killed and 30,000 were taken 

 prisoners, and all the baggage and artillery fell into the victors' 

 hands. " The Swedes will teach us how to conquer them," 

 said Peter after the battle, and at once he took steps for bring- 

 ing another army into the field. Charles XII. continued on a 

 long series of victories. Poles, Saxons, and Eussians melted 

 away before him ; the King of Poland was dethroned at his 

 dictation, and a nominee of his own raised in his stead; the 

 Emperor of Germany had to concede certain things not by any 

 means to his taste ; and all Europe trembled when the King of 

 Sweden marched. This went on from 1702 to 1706, and then 

 the czar, having a large army at his back, thought he might 

 seek peace with honour. But Charles declared that he would 

 not talk of peace till he reached Moscow, which he proposed 

 to burn. Like another invader (Napoleon I.), he found the 

 Russians prepared to do anything rather than see their capital 

 in an enemy's hand. Peter devastated the country, harassed 

 the march of the Swedes, cut off the discontented Cossacks, 

 who were in secret alliance with Charles, and in other ways 

 hindered his operations. Finally, at Pultowa which fortress, 

 in the Ukraine, Charles was besieging the czar came up with 

 his enemies ; a bloody battle ensued, in which the most despe- 

 rate valour was shown, but the Swedes were utterly routed 

 8,000 were slain and 18,000 captured. Charles was obliged 

 to seek refuge in Turkey, where he employed himself in trying 

 to promote the anger of the Turks against the Eussians, but 

 he was never thenceforth the thorn he had been in the side of 

 the czar. 



Peter, freed from external troubles, again turned his atten- 

 tion to home affairs. St. Petersburg was finished, and the other 

 great works were brought to a successful termination ; vast 

 strides were rapidly made in the improvement of all public 

 institutions ; and the czar had the happiness before his death 

 to find by many infallible signs that he was really looked upon 

 as the father of his country. 



The Eussia which he left in 1725 was so radically altered 

 in character to the Eussia to which he had succeeded, that it 

 could flourish and be prosperous under the hand of a woman, 

 Peter's widow, who succeeded him as Catherine I. The height 

 to which Catherine II. and successive emperors have raised it 

 is matter rather of general history than for an historic sketch. 



LESSONS IN ALGEBRA. XXVI. 



ADDITION OF POWERS. 



IT is obvious that powers may be added, like other quantities, 

 by writing them one after anotlier, with their signs. 



EXAMPLES. The sum of a, 3 and b 9 is o, 3 -f t 2 ; and the sum of 

 a 2 - 6" and h b - d* is a" - b n -f 7i s - d 4 . 



The same powers of the same letters are like quantities, hence 

 their co-efficients may be added or subtracted. 



EXAMPLE. Thus the sum of 2a 2 and 3a 2 is So, 2 . 



But powers of different letters, and different powers of the 

 same letter, are unlike quantities; hence they can be added only 

 by writing them down with their signs. 



EXAMPLE. The sum of a" and a 3 is a 2 -f a 3 . 



It is evident that the square of a, and the cube of a, are 

 neither twice the square of a, nor twice the cube of a. 



EXAMPLE. The sum of a 3 b n and 3a 5 6 6 is a& 3n -f 3a 5 6 6 . 



From the preceding principles we deduce the following 



GENERAL RULE FOR ADDING POWERS. 



If tJie powers are like quantities, add their co-efficients, and 

 to the sum annex the common letter or letters with their given 

 indices. 



If the powers are unlike quantities, they must be added by 

 writing them one after another, without altering their signs. 



EXERCISE 42. 



1. To -3x^5 add -2ci/ r . I 



2. To 3b add 66-. 



3. To 3oV add -7a*i;. 



5. To 3(a + j/) add 4(a + y)'. 



S. Add 5i'(a - 6) 3 + *(a - b) 3 to 



2r(a - b) 3 + 10.v(a - 6)3. 

 7. Add 3(,r + y)* + 5a 3 - 4(x + y)* 



to 10n 3 + 6U- + )*. 



8. Add 5a 2 6c3, 3a"6c3, 

 2a"6c 3 . 



9. Add 



and 



a 3 b s + x'y* + a 3 b 3 an( j 



x*y* + a*6 a . 



10. Add 3a 3 + be 3 + 5o + 2bc a and 

 a 3 + 56c" to 6a s + 26c 3 . 



11. Add -Kxy-cm)', 3(.ry-cm) B , 



1 (*!/ cm) *, and J (xy cm) * . 



SUBTRACTION OP POWERS. 



EULE. Subtraction of powers is performed in the same 

 manner as addition, except that the signs of the subtrahend, 

 must be changed as in simple subtraction. 



EXAMPLE. From 2a 4 take - 6a 4 . Ans. 8a 4 . 



EXERCISE 43. 



1. From -36 take 46". 



2. From 3?i 3 b take 4Jt 3 b'. 



3. From a3b take a3b". 



4. From 5(a - ?i) take 2(o - h) a . 



5. From 6a(a + b)* take a(a + b)*. 



6. From 17a a r 1 + 5xy take 12a a 2 4aT/ 3 . 



7. From 3a3(b 3 - 8)3 take a 3 (b 3 - 8)3. 



8. From 5(x 3 + y*)3 - 3(a 3 - 6 3 ) 5 take - 3(a - b) 5 + 4(* s + g)3. 



9. From a"b+ .tV take a 5 6- x*y 3 . 



10. From 2x(a - 6) s + 3(a - 6) 3 ;take x(a - b) 3 + 3(a - b)3. 



11. From l(x + ij) + J(a + b) s take l(x + v) 3 + f(a + by*. 



MULTIPLICATION OF POWERS. 



Powers may be multiplied, like other quantities, by writing 

 the factors one after another, either with or without the sign of 

 multiplication between them. 



EXAMPLES. The product of a 3 into 6 2 , is a 3 6 2 ; and x* into 

 a m , is a m a; 3 . 



If the quantities to be multiplied are poiuers of the same root, 

 instead of writing the factors one after another, as in the last 

 article, we may add their exponents, and the sum placed at the 

 right hand of the root will be the product required. 



The reason of this operation may be illustrated thus : 



a 2 X a 3 is a-a 3 ; but a, 2 = aa, and a 3 = aaa ; and aa X aaa 

 = aaaaa = a s . The sum of the exponents 2 + 3 is also 5 ; so 

 d m X d n = d m + n . 



N.B. The same principles hold true in all other powers of 

 the same root. 



Hence we deduce the following 



GENERAL RULES FOR MULTIPLYING POWERS. 



Powers of the same root may be multiplied by adding their 

 exponents. 



If the powers have co-efficients, these must be multiplied to- 

 gether, and their product prefixed to the common letter or letters. 



Powers of different roots are multiplied by writing them one 

 after anotlier, either with or ivithout the sign of multiplication 

 between them. 



EXAMPLES. Thus a 2 X a 6 - a 2 + 6 = a 6 ; and x 3 X x 2 X x 



The rule is equally applicable to powers whose exponents are 

 negative; i.e., to reciprocal powers. 



EXAMPLES. 

 Thus a- 2 X a- 3 = a- 5 . 



That is, x -- = - __ 

 aa aaa aaaaa 



If a -f- b be multiplied into a b, the product will be a 2 b" ; 

 that is 



The product of the sum and difference of two quantities is 

 equal to the difference of their squares. 



This is an instance of the facility with which general truths 

 are demonstrated in algebra. 



If the sum and difference of the squares be multiplied, the 

 product will be equal to the difference of the fourth powers ; 

 that is, (a 2 + &) X (a 2 - 6 2 ) = (a* - b 4 ). 



EXERCISE 44. 



1. Multiply h*b* into a 4 . 



2. Multiply 3a"y 3 into 2jc. 



3. Multiply dhV into 4by. 



4. Multiply a 3 !/ 3 !/ 3 into a 3 6 a y. 



5. Multiply 4a into 2a. 



6. Multiply 3z* into 2z. 



7. Multiply 6'y 3 into b*y. 



8. Multiply a*b 3 y* into c^b'y. 



9. Multiply (b + h - y) 



into 



10. Multiply nfl+x*y+xy* +y 3 into 



x-y. 



11. Multiply 43!'^ + 3xy - 1 into 



2.v J - x. 



12. Multiply x* + x-5 into 2i + 



x + I. 



13. Multiply y* into y m into y*. 



14. Multiply a'" into a~ 3 into a- 8 . 



15. Multiply <r 3 into a- * into a -J . 



16. Multiply a'" into a into a"". 



17. Multiply j/" 3 into j/" into 



-y-"y 3 . 



18. Multiply (a y) into (a + y). 



19. Multiply (a 2 y*) into (a 3 + iy 2 ). 



20. Multiply (a*- y*) into (a*+ y*). 



21. Multiply a 3 + a*+o 6 into a 3 1. 



22. Multiply 3a(a;' ! - y 3 ) 3 into 



2<t(a, J > -y 3 )* 



23. Multiply J(a + b 3 ; 8 into 



ifa 3 + b)". 



24. Multiply a 3 b= into o> + b 3 . 



25. Multiply x 3 + x'y + xy"+ y'into 



x + y. 



26. Multiply a*- 2a s b+ 4a 2 62- 8ab 3 



+ 166* into a + 26. 



27. Multiply a 2 + 6 into a*- 8. 



