156 



THE POPULAR EDUCATOR 



, the 



(1.) Reduce the indices to a common denominator. 

 (2.) Raise each quantity to the power ex}>ressed by the nume- 

 rator of its reduced index. 



(3.) Take tlie root denoted by the common denominator. 



EXAMPLES. 

 Eeduce a* and b 6 to a common index. 



1st. The indices J and reduced to a common denominator 



are ^ and $. 

 3nd. The quantities a and b raised to the powers expressed 



by the two numerators are a 3 and b-. 

 3rd. The root denoted by the common denominator is the 



1 -- 



i^th. The answer, then, is (a 3 ) 1 '- and (6-) 12 . 

 The two quantities are thus reduced to a common index, 

 without any alteration of their values. 



For or = a", which = (a 3 ) 12 



I m i 



And universally, a u = a"" 1 = (a m ) mil - 



1 a 



Eeduce a. 2 and (foe) 3 to a common index. 



Here a* and (bxft = a* and (62:)*, or (a 3 )* and (6 4 a: 4 )5. 



CASE III. To reduce a quantity to one with a given index. 



Divide the index of the quantity by the given index, place the 

 quotient over the quantity, and set the given index over the 

 whole. 



This is merely resolving the original index into two factors. 



EXAMPLES. 

 Reduce a 5 to one with the index -J. 



1-1-1 1 V - 2 1 



o !i B^i a 3- 

 This is the index to be placed over a, which then becomes 



i i i 



a 3 ; and the given index set over this, makes it (a 3 ) 

 answer. 

 Eeduce a 2 and a; 2 to others with the common index ^ 



2 -j- J = 2 X 3 = 6, the first index. } 

 1 -T- J = 3 X3 = |, the second index, j 



1 91 



Therefore (a 6 ) 3 and (x-) J are the quantities required. 

 EXERCISE 48. 



1. Eeduce 4 to the form of the i 10. Eeduce 2* and 3*. 



cube root. j n. Eeduce (a + b) and (* 



2. Eeduce 3a to the form of the | 12 -R^,,^ } nilil h } 



4th root. 



3. Eeduce |ab to the form of the 



square root. 



4. Eeduce 3 x (a x) to the form 



of the cube root. 



5. Eeduce a 2 to the form of the 



cube root. 



6. Eeduce aW to the form of the 



square root. 



7. Eeduce a to the form of the 



nth root. 



18. Eeduce a and b* to others 



with the common index 7. 



19. Eeduce a 2 , b*, and c* to others 



with the common index T V 

 CASE IV. To reduce a radical quantity to its most simple 



terms ; i.e., to remove a factor from under the radical sign. 

 Resolve the quantity into two factors, one of which is an exact 



power of the same name with the root. Find the root of this 



power, and prefix it to the other factor, ivith the radical sign 



betiveen them. 



This rule is founded on the principle that the root of the 

 product of two factors is equal to the product of their roots. 



It will generally be best to resolve the radical quantity into 

 such factors, that one of them shall be the greatest power which 

 will divide the quantity without a remainder. 



N.B. If there is no exact power which will divide the quantity, 

 the deduction cannot be made. 



EXAMPLES. 

 Eemove a factor from A/ 8. 



The greatest square which will divide S is 4. We may then 

 resolve 8 into the factors 4 and 2 ; for 4 X 2 = 8. 



8. Eeduce a 3 and b" to a common 



index. 



9. Eeduce x* and i/ 1 ". 



13. Eeduce x$ and 5. 



14. Eeduce 4 J and 3^ to others 



with the common index J. 



15. Eeduce a: 3 and i/Ho others with 



the common index |. 



16. Eeduce a 1 and b 3 to others 



with the common index J. 



17. Eeduce c and d' to others 



with the common index J. 



The root of this product is equal to the product of the roots 



of its factors ; that is, V 8 = V 4 X V 2. 

 But A/ 4 = 2. Instead of A/4, therefore, we may substitute 



its equal 2. We then have 2 X V 2, or 2 V 2. for the 



answer. 



Eeduce +/a' 2 x. Ans. A/a 2 X A/X = a X A/ = 



CASE V. To introduce a co-efficient of a radical quantity 

 under the radical sign. 



Raise the co-efficient to a power of tJie same name as the 

 radical part, then place it as a factor under the radical sign. 



EXAMPLES. n 



Thus, a n V b = n V a n b. For a = n / a n , or a n ; and 

 X n A/6 = n A/~o"6. 

 Eeduce a(x - 6)* to the form of a radical. 



a(x - b)$ = l/a 3 (x - b) = (a 3 x - a 3 6)^. 

 EXERCISE 49. 



1. Eeduce 



form. 



2. Eeduce VSlPc 



3. Eeduce i/ 



^ 



4. Eeduce " \/ a"b. 



5. Eeduce (a" a^o 



6. Eeduce (54a6b)i 



7. Eeduce </9Sa?x. 



to its simplest 



8. Eeduce V a 3 + a 3 b 2 . 



9. Eeduce 2ab(2a* 2 )i 



10. Eeduce -f-^-V. 



6 v a s + b 3 ' 



11. Eeduce 2 v/2. 



12. Eeduce 46 s Jc. 



13. Eeduce 5 ,/6 to a simple radical 



form. 



14. Eeduce \-Joa to a simple 



radical form. 



15. Eeduce 5* and 6* to others 



with the common index -J. 



16. Eeduce a 2 and a" to others 



with the common index J. 



17. Eeducs V98 to its simplest 



form. 



18. Eeduce V243 to its simplest 



form. 



19. Eeduce 3^/54 to its simplest 



form. 



20. Eeduce 7V80 to its simplest 



form. 



21. Eeduce 9 3^/81 to its simplest 



form. 



22. Eeduce V*+ a* 2 to its simplest 



form. 



23. Eeduce </198a?x to its simplest 



form. 



24. Eeduce 



V J a*** to 

 simplest form. 



its 



ADDITION OF EADICAL QUANTITIES. 



It may be proper to remark, that the rules for addition, sub- 

 traction, multiplication, and division of radical quantities 

 depend on the same principles, and are expressed in nearly the 

 same language, as those for addition, subtraction, multiplica- 

 tion, and division of powers. So also the rules for involution 

 and evolution of radicals are similar to those for involution 

 and evolution of powers. Hence, if the learner has made him- 

 self thoroughly acquainted with the principles and operations 

 relating to powers, he has substantially acquired those pertain- 

 ing to radical quantities, and will find no difficulty in under- 

 standing and applying them. 



When radical quantities have the same radical part, and are 

 under the same radical sign or index, they are like quantities. 

 Hence their rational parts or coefficients may be added in the 

 same manner as rational quantities, and the sum prefixed to 

 the radical part. 



Thus, 2 Vb + 3 A/5 = 5 A/6. 



If the radical parts are originally different, they may some- 

 times be made alike, by the rules for reduction of radical 

 quantities. 



EXAMPLE. Add A/8 to A/50. 



Here the radical parts are not the same ; but by reduction, 

 A/8 = 2-V/2, and A/50 = 5A/2; and 2A/2 + 5A/2 

 = 7 A/2. Ans. 



EXERCISE 50. 



6. Add V16b to V46. 



7. Add Va 2 * to Jb*x. 



8. Add (36a*y)* to (25y)*. 



9. Add 



1. Add " -Jay to 2 -J ay. 



2. Add-2v/ato5Va. 



3. Add 4(2+ h)^to3(a: + 7i)^. 



4. Add 7Mi if 5bh. 



5. Add y */b h to a *Jb h. 



If the radical parts, after reduction, are different, or have 

 different exponents, then the quantities, being unlike, can be 

 added only by writing them one after the other with their 

 signs. 



