Rationalize the denominator of 



Radicals. 49 



4+^21 



v^7— ^3— V 2 



%/i8 

 7. Rationalize the denominator of 



2 \/ 2 — v^ 3 

 (S'. Rationalize the denominator of — 



3 + 2\/2 



^. What relation must hold between a and x in order that 



a 4- ^'^-^' 



%/ 2 — ^'^ + '^S 

 /o. Rationalize the denominator of - _— -_ -—. 



V2-Hv^34-^5 



//. Compute the value of the following to three places of 



decimals, having first reduced it to its simplest form : 



^3 + v/^-^5^v/5 



^Z^VlW^-s/ 



12. Prove , ::--^^-_ ^ -^:::^:^_ -{- ~ r = 7.X^ . 



Vx'-^- 1 — v/.r"— I v/.r^-f i -j- Vx"— i 



30. Theorem. If a, b,p, q are co trim ensur able ajid ^ b and^q 

 incommensurable, arid if a-\-\^ b =p + ^q , then a=p and^^b = >/ q . 



\i a does not equal/, suppose a=p + d. Substitute this value 

 for a in the given equation, and we have 



p-\-d-{-s/b=p+\^q 

 or d-\-\^b=s/q 



squaring both sides 



whence 



d'-^2ds/b-^b^q 

 2d 



That is, an incommensurable quantity equals a commensurable, 

 which is absurd. Therefore a cannot differ from p. And \i'a=^p, 

 ^ b must equal >/q . 



A— 6 



