simplify each expression write your answers with positive exponents

(a^1/2 b) 1/2 (ab^1/2)

To simplify the expression (a^(1/2) * b)^(1/2) * (ab^(1/2)) and write the answer with positive exponents, we can follow these steps:

Step 1: Simplify the expression inside the first parentheses.
The expression inside the first parentheses is a^(1/2) * b. To simplify this, we apply the rule of exponents that says when you multiply two terms with the same base, you add their exponents.
So, a^(1/2) * b can be rewritten as a^(1/2) * b^1.

Step 2: Simplify the expression inside the second parentheses.
The expression inside the second parentheses is ab^(1/2).

Step 3: Multiply the simplified expressions from steps 1 and 2.
(a^(1/2) * b^1) * (ab^(1/2))

Using the rule of exponents again, we combine the variables with the same base, 'a', and add their exponents. We do the same for the variables with the same base, 'b', and add their exponents.

a^(1/2) * a^1 = a^(1/2 + 1) = a^(3/2)
b^1 * b^(1/2) = b^(1 + 1/2) = b^(3/2)

So, the expression simplifies to:
(a^(1/2) * b)^(1/2) * (ab^(1/2)) = a^(3/2) * b^(3/2)

Therefore, the simplified expression with positive exponents is a^(3/2) * b^(3/2).