Simplify each expression using the rules of exponents and examine the steps you are taking.Incorporate the following five math vocabulary words into your discussion.

Product rule
Principal rooot
Quotient rule
Reciprocal
nth root

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

Well, well, well! We've got ourselves an interesting expression here. Let's break it down and simplify it using our friend, the rules of exponents.

First, we see that we have a product inside a principal root. According to the product rule, we can split this up into two separate principal roots. So, let's rewrite the expression as (a^1/2 b)^1/2 * (ab^1/2).

Next, we can simplify each of these principal roots separately. For the first principal root, we have (a^1/2 b)^1/2. According to the rule of taking a principal root of a power, we can multiply the exponents. So, we have a^(1/2 * 1/2) * b^(1/2 * 1/2), which simplifies to a^1/4 * b^1/4.

For the second principal root, we have (ab^1/2). This stays the same because it is already as simplified as it can get.

Now, let's put everything together. We have a^1/4 * b^1/4 * (ab^1/2). To multiply variables with the same base, we can add the exponents. So, we have a^(1/4 + 1/2) * b^(1/4 + 1/2), which simplifies to a^(3/4) * b^(3/4).

And voila! We have simplified the expression using the rules of exponents. Remember, it's always important to understand the steps we're taking and how each rule applies. Keep practicing those math vocab words ("product rule," "principal root," "quotient rule," "reciprocal," and "nth root") and soon enough, you'll be a math wiz!