IN LLO1BR4, 



HI 



K:,; :: t|| 



I. Diride v'tf^J by 



':) (u s + ux)' by u 

 . > (aS/i)"- by (ur,)-. 

 U. (aV)* by (ay)*. 

 i3a)*. 

 !o (<u;)*by 



! by ,(ai) 1 . 



10. Diride 



11. 



13. Divide 



13. Divide 



n 



U. Diride6v/ry by 

 6. Divide a by a' 



EXERCISE 58. EXAMPHS FOE PBACTICI. 



7. Divide VaM by I.'. 



8. Diride OV433 by3V4. 



9. Diride -z* by J-t*. 



10. Divide \/7 by * ^7. 



11. Diride 66 V6* by lO**^**. 



12. Diride 2 V by 3 v '. 



. Divide (64-)t 



by (i 



lo2Vbo by 3/oc. 

 videVS by a, 1 .'. 

 vide 10 ^108 by 5*^84. 



4. Diride V32 by V->. 



5. Divide V30 by V5. 

 . Diride o u Va by a*. 



INVOLUTION OP RADICAL QUANTITIES. 

 To involve a radical quantity to any required power, 



Multii>t>j tli>- imlexofthe root into the index of the power to 

 which i( is to be n 



EXAMPLE. Thus the square of a* = a* * = 

 a$ X a* = a* 



For 



A root is raised to a power / th<s same name by removinj the 

 index or radical t 



N.B. When tho radical quantities have rational co-efficients, 

 these must bo involved by actual multiplication. 



Thus tho cube of \/b -+ a;, is I -f x. 



And tho nth power of (a - 1/), is a - y. 



The square of a "Var, is a 2n A/I". 



For a n */x X a n */x = a 2 n \/x*. 



But if the radical quantities are connected with others by tho 

 signs + and - , they must be involved by a multiplication of tho 

 several terms. 



EXAMPLE. Eequired tho square of a + */y and of a - */y. 



a + Vy a Vy 



a + Vy a Vy 



a 2 + aVy a 2 



y 



a 2 2a*/y+ y. 

 EXERCISE 57. 



1. Required the cube of a*. 



2. Required the nth power of ai. 



3. Required the 5th power of a^y*. 



4. Required the cube of B 



5. Required the square of a***. 

 0. Required the cube of a*. 



7. Required the nth. power of a'. 

 0. Required the nth power of 

 o m *. 



9. Required the square of a v/x ;/. 



10. Required the cube of 3a 3 Vy. 



11. Required the cube of o bjl. 



12. Required the cube of o ,/6. 



13. Required the 4th power of Jd. 

 li. Required the 4th power of 



15. Required the 6th power of 



f/a -r b. 



EVOLUTION OF RADICAL QUANTITIES. 



Tho operation for finding the root of a quantity which is 

 already a root, is the same as in other cases of evolution. 

 Hence we derive the following 



RULE FOR THE EVOLUTION OF RADICALS. 



Divide the fractional index of the quantity by the number 

 expressing the root to be found. Or, 



Place the radical si<jn belonging to the required root over the 

 given quantity. 



If the quantities have rational co-efficients, the root of thete 

 must be yctracted and placed before the radical sign or quantity. 



EXAMPLE. Thus tho square root of a', is a' ~ " = o^. 

 From tho preceding rules it will be perceived that jxww* and 



- 





EZMOUI U. 



L Jladtbaeaaerootaf efo)*- 



S. 



1 Find tb Uh root of /4 yfc. 



4. FlaattMnhnMtaf mv*. 



5. Find th* 4th root <rf . 

 0. FiudtUflttrootof ,, 



7. Fi4Uithru., 

 9. Find UMoate root uf- 

 9. Ftedtbo tqiur* root o< 

 10. Had the (qau* root ot 



11. Find th* 5th root of 



13. Find UM aqaar* root of a* 



IS. Bdao to tha form of UM 

 fehroot. 



14. Reduce -3j to the form of UM 



. '.. ..: I 



15. Reduce * and a* to ft oommoa 



index. 



10. R*dueo 4* and 5* to a common 



i:. h x. 



Jf. Dirid5^Uby VI 

 80. Drfid4VSbr VS. 

 31. FtadUMeaUolJ^l. 

 Jt. Fiad UM qwn of J -f VS. 



53. Find UM 4th powvr ofl 4- /I 



54, Had UM omte of S 4- V*. 



of UM 



W aball 



REDUCTION OF EQUATIONS BT IXVOLUTIOK. 



In an equation, the letter which tipliam UM onkuown 

 quantity is ometinat found under a radical y. W 10*7 

 have v'x = a. 



To clear thia of the radical ngn, let each 

 equation bo aqoared, that u, multiplied into itaelf. 

 then have Vx X v'x = oxi. Or, * = a*. 



The equality of tho sides in not affected by tida 

 because each is only multiplied into italf, that u, equal 

 tities are multiplied into equal qoantttiea. 



The same principle u applicable to any root whatever. If 

 " -/x = a, then z = a n . For a root u raJMd to a power of the 

 same name, by removing the index or radical sign. 



Hence, to reduce an equation, when the onknoira cpmmtity is 

 under a radical sign, 



Involve both tide* to a power of the tamo aaM <u tkt not 

 expressed by the radical sign. 



N.B. It will generally be expedient to make the 

 transpositions, and to clear the equation of fraction*, 

 involving the quantities ; so that all those which are not 

 tho radical sign may stand on one aide of the equation. 



EXAKPLK*. 



Eeduco the equation, 

 Transposing +4, 

 Involving both aides, 



Bednoe the equation, 

 By transposition, 

 By involution, 



/ 4-4=9. 

 \ f x = 9 4 = 5. 

 r = 5* = 25. 



a -* v'* b = <. 



* v'x = d + b a. 

 r z= (d -f 6 - ). 



EXERCISE 59. 



1. Reduce the equation V* i- I f 10. Bedaea y (-). 



2. Reduce the equation 4 4-3 'z- 



3. Reduce the equation * V(4+ 7) 



+ 4-1& 



4. Reduce % 'ii -10 + 4-14. 



5. Reduce Vf " 8 - 



6. Reduce (2* + 3)* + 48. 



7. Reduce V 13 + * - 2 -r 



8. Reduce V(S* + I) + 5 - 10. 



9. Reduce /(* -f ) -e- </( -I- b). 



9 AVy (o E*trcite$ 53, 54 tcOl t ^M ia MT/ IMOTI. 



