Suppose we have the expression (16/81)^-1/2. Remember what we do with expressions with negative exponents? We turn the negative exponent into a positive exponent, and we invert the fraction so that we have its reciprocal. Therefore, our (16/81)^-1/2 will become (81/16)^1/2. Next, we must hark back to the laws of exponents we went over in a previous article. We distribute this exponent to both the numerator and the denominator within the parenthesis. Therefore, our (81/16)^1/2 becomes (81^1/2 / 16^1/2).
Now when we have fractional exponents, we take the denominator, and use that denominator as the index of the radical expression into which we convert the expression. Therefore, 81^1/2 becomes 81 with a square root sign over it (81 is our radicand) with 2 as the index. Of course, we would not write the index in this case, because it is taken for granted that when we have a radical expression, a 2 is implied. The square of 81 is 9 and the square of 16 is 4. Therefore, our answer is 9/4.