#r^(1/2)/r^(-1/4)=r^(1/2)-:r^(-1/4) #
Here, since the #color(red)("bases are same")#, and we have to find the #color(red)("divide the terms")#, we can simply #color(red)("SUBTRACT THE POWERS")#:
#r^(1/2)-:r^(1/4) = r^(1/2-(-1/4)) #
#= r^(1/2+1/4) #
#=r^((2+1)/4)#
#=r^(3/4)#
ALTERNATELY
We know that:
#x^-n=1/x^n#
Therefore,
#1/x^-n=1/(1/x^n)=x^n#
#=>1/r^(-1/4) = r^(1/4)#
The given expression can be written as:
#r^(1/2)xxr^(1/4)#
Here, since the #color(red)("bases are same")#, and we have to find the #color(red)("multiply the terms")#, we can simply #color(red)("ADD THE POWERS")#:
#r^(1/2)xxr^(1/4)=r^(1/2+1/4)#
#=r^((2+1)/4)#
#=r^(3/4)#