Final answer:
To show that (a/b)^(a/b) = a^(a/b-1), we can use the property of exponents. To prove that b = 2 when a = 2b, we substitute a = 2b into the given equation (a^b = b^a).
Step-by-step explanation:
To show that (a/b)^(a/b) = a^(a/b-1), we can use the property of exponents that states (a^b)^c = a^(b*c). Let's rewrite the left side of the equation as ((a/b)^a/b)^1. Using the property mentioned, we can rewrite this as (a/b)^(a/b * 1), which simplifies to (a/b)^(a/b).
To prove that b = 2 when a = 2b, we substitute a = 2b into the given equation (a^b = b^a). We get (2b)^b = b^(2b). Simplifying this, we get 2^b * b^b = b^(2b). Since 2^b = b^2 (from b = 2), we can substitute and rewrite the equation as b^2 * b^b = b^(2b). Combining the exponents, we get b^(2+b) = b^(2b). Therefore, 2+b = 2b, which simplifies to b = 2.