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)