LESSONS IN AUiKHKA. 



MET 





1. Required the 3rd power of 3*. 



, nired the 4th pow. 

 8. Required the 7th pow. 

 4. What is the 3rd power of 



it is the nth power of 

 ady? 

 li. What is tho -ith power of 



t is the 3rd power of 46 7 

 8. What is tho nth power of 



11. Imd the 3rd power of 4s*. 

 nh power of ita x 



IS. Required the 5th power of 

 (a + b). 



14. Required the 2nd power of 



(a+b)n. 



15. Required the nth power of 



(*-y)m. 



16. Required the nth power of 



(* **" v) % - 



17. Required the 2nd power of 



(o x b 1 ). 



18. Find the 3rd power of 



0. What is the 3rd power of 



3m 

 10. Find the 4th power of n'b 1 . 



183. A FRACTION is raised to a power by involving both the 

 numerator and Uw denominator to the powei required. 



EXAMPLE. Find the square, of -. 



b 



By the rule for the multiplication of fractions we have a X - 

 aa a* 66 



~~ bb "&' 



184. A compound quantity, consisting of terms connected by 

 4- and , is involved by an actual multiplication of its several 



EXAMPLE. Find the 2nd, 3rd, and 4th powers of a -f b. 



Here, (a 4- b) 1 = a 4- b, the first power ; 



a 4- b 



a 2 4-ab 

 + ob 4- b 2 



the second power ; 



a* + 2a 2 b-fab 



+ o'b + 2ab + *>* 



3ab 2 4- b 3 , . the third power ; 



a 4 + 3a s b + 3a 2 b 2 + ab 



+ ab + 3a 2 b 2 + 3ab* + b 



power. 



(a 4- b) = o + 4ab 4- 6a*b 2 4- 4ab s 4- b, the fourth 



EXERCISE 32. 



1. Find the 2nd, 3rd, and nth powers of - 

 2rr> 



2. Find the cube of 





3. Find the nth power of _. 



4. Find the square of ~ a>X (d + m \ 



(x -r 1)' 



5. Find the square of a b. 



6. Find the cube of a + 1. 



7. Find the square of a + b + 7i. 



8. Required the square of a + 2d + 3. 



9. Required the 4th power of b -h 2. 



10. Required the 5th power of x + 1. 



11. Required the 6th power of 1 b. 



185. The squares of binomial and residual quantities occur so 

 frequently in algebraic processes, that it is important to make 

 them familiar. Thus, 



If we multiply a 4- h into itself, and also a h into itself, we 



a+h a h 



o> + h a h 



o*4-a7i a 2 ah 



+ ah 4- h- - 



a*+2ah + K> a t 2ah + h t . 



Here it will be seen, that in each case the first and last 

 terms are the squares of a and h ; and that the middle term is 

 twice the product of a by h. Hence the squares of binomial and 

 residual quantities, without multiplying each of the terms 

 separately, may be found by the following rule : 



(1.) The square of a BINOMIAL, the terms of which are both 

 positive, is equal to the squares of the Jirst and last terms, plus 

 f <ri', v flic product of the two terms. 



(2.) The square of a RESIDUAL quantity is equal to the squares 

 of the first and last terms, minus twice the product of the two terms. 



:;:;. 



1. Find the square of U + fc. 14. Find the square of 6 y + 3. 

 8. Find the square of k + 1. ft. Find the squire of M - k. 



3. Find the square of at> + L | 0. Find UM square at* L 



186. For many purpose* it will bo sufficient to eipmn th 

 powers of compound quantities by exyoiunit witbooi aa rrfrTfJ 

 multiplication. 



EXAMPLE*). 



1. Find the square of a + 6. Ant. (a + b)'. 



2. Find the nth power of be + 8 + r. An*.(bc + 8 + ). 

 In cases of thin kind, all the term* of which the compound 



quantity consist* mast be included in the parentheoi. 



187. But if the root consist* of several factors, the parstv 

 thesis used in expressing the power may either extend over the 

 whole, or may be applied to each of the factors separately, M 

 convenience may require. 



Thus the square of (a -f b) X (c + d), is either 



