28o ALGEBRA. 



This rule is equally applicable, when the exponents of 

 any roots of the same quantity are fractional. 



Thus, the product of ^z* multiplied into a' is «*X«*±i 

 In like manner, a;"^X^^Xa;^ z: a;^"*'^'*'^ zz x^ :zzx' 



Hence it appears, that, if a surd square root be multi- 

 pheH ifito*itself, the prdcHlct will be rational ; and if a 

 surd cube root be multiplied into itself, and that product 

 into the same root, the product is rational. And, in gen- 

 eral, when the sum of the numerators of the exponents 

 is divisible by the common denominatorj \^ithOut a re- 

 mainder, the product will be rational. 



X. ' 5_i_? 111. 8 

 Thus, a^Xa^ZZa^'^'^ZZa ^ ZZa^ZZ:a\ 



8 



Here the quantity a^ is reduced to a^, by actually di- 

 viding 8, the numerator of the exponent, by its denomi- 

 nator 4 ; and the sum of the exponents, considered mere- 

 ly as vulgar fractions, is ■^'{'^zz^zz:2. 



When the sum of the numerators and the denominator 

 of the exponents adjuit of a common divisor greater than 

 unity, then the exponent of the product may always be 

 reduced, like a vulgar fraction, to lower terms, retaining 

 still the same value. 



Thus, x^Xx'^zzx^-ixK 



Compound surds of the same quantity are multiplied in 

 the same manner as simple ones. 



J. - X a , I 



Thus, a + x\* X a + xl"" = a+xl^ = c+«| = a+x ; 



