Final answer:
To express (81)^¹/²⁵ in simplest radical form, rewrite 81 as 9^2 and use the laws of exponents to obtain (9^2)^¹/²⁵ = 9^(1/16), which is the 16th root of 9 and cannot be simplified further.
Step-by-step explanation:
To express (81)^¹/²⁵ in simplest radical form, let's first simplify the expression inside the radical. Since 81 is a perfect square, we know that 81 = 9^2. Now, we can rewrite the original expression using this equivalent expression for 81 and apply the laws of exponents:
(81)^¹/²⁵ = (9^2)^¹/²⁵
By the laws of exponents, when we raise a power to another power, we multiply the exponents. Therefore, we multiply 2 by 1/2^5, which simplifies to:
(9^2)^¹/²⁵ = 9^(2 × 1/32) = 9^(1/16)
Now, the expression 9^(1/16) means we are looking for the 16th root of 9. Since there is no simpler radical form for the 16th root of 9, we leave our answer as:
9^(1/16), which is the 16th root of 9.