Radicals. ' 47 



(x-jKx"--\-x"-y-j-x"-y^-{- . . .y'-^)=x"-y\ 

 by substitution, I, Art. 26. But 



which is rational. Therefore (i) is the rationalizing factor for 



a" — c" . 



s t 



(b) To rationalize the form a" -\- c" . 



— ~ ± L 



As before, let a"=x and c"=j'; whence a" -\- c" =x-{-j\ 



Multiply x-i-j by * 



x"-'—x"-'j+x"-y— . . . =by-'. (2) 



The product is 



^ -^^ -^ ' -^ -' ^ ( -r"+y if ^Msodd, 



by I, Arts. 27, 28. But 



r i.1" r 



Both of these results are rational ; therefore (2) is the rational- 

 izing factor. 



27. Examples. 



2 L 



.2 



1. Rationalize d'^—r-. 

 With a common denomininator for the exponents this becomes 



d^—f^\ whence ;z = 6, ^=4, ^=3 ; then ^-=^«, J/=r^ 

 . (x—y)(x^' -^x^y-Vx^y"- -{-x^^ -\-xy* -\-y^ ) 



f 2 J_l f 10 8 1 0. 2_ 4 ;L 2 4 41 ^ 



= d^ — r^, which is rational. 



2. Rationalize 6 + 3 ^ k,. 



With a common denominator for the fractional exponents this 



becomes 6* + (3^ X5)*; whence ?^=4, ^=4, ^=1 : then a-=6'* and 



i 

 1/= (34x5)**. Therefore 



