Final answer:
The expression ((a⁴b⁻²)/(a⁸b⁶))¹/² simplifies to 1/(a²b³) when we subtract the exponents of like bases, convert negative exponents to fractions, and then take the square root of both parts.
Step-by-step explanation:
To simplify the expression ((a⁴b⁻²)/(a⁸b⁶))¹/², we need to apply the properties of exponents. First, we'll divide the powers by subtracting the exponents of like bases (since the bases a and b are the same in both the numerator and the denominator, and we're dividing one by the other).
For the base a: a⁴ / a⁸ = a^{(4-8)} = a⁻⁴.
For the base b: b⁻² / b⁶ = b^{(-2-6)} = b⁻⁸.
The expression now is (a⁻⁴b⁻⁸)¹/².
Since negative exponents denote a division, a⁻⁴ means 1/a⁴ and b⁻⁸ means 1/b⁸. Now, raise both fractions to the power of ¹/² (which is the same as taking the square root).
Therefore, (a⁻⁴b⁻⁸)¹/² becomes (1/a⁴)¹/² * (1/b⁸)¹/², which further simplifies to 1/(a²b³) since (1/a⁴)¹/² is 1/a² and (1/b⁸)¹/² is 1/b³.
So, the simplified expression without negative exponents is 1/(a²b³).