The square of $a$ and the square root of $b$ vary inversely. If $a=2$ when $b=81$, then find $b$ when $ab=48$.

a^2 = k / sqrt (b)   where k is the proportionality constant

So

2^2  = k / sqrt (81)

4  = k / 9

4 * 9  = k =  36

So

If  ab =  48  then  a =  48/ b

Then

(48/b)^2  =  36 / sqrt (b)

2304 / b^2 =  36 / sqrt (b)

2304 / 36  =  b^2 / sqrt (b)

2304/36 = b^2 / b^(1/2)

2304/36  = b^(3/2)      

(2304/36)^(2/3)  = (64)^(2/3)  = (64^(1/3))^2  = 4^2  =   b  =  16