need help understanding exponents using the product rule/ Simplifying Radicals.

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

assume your x means * or "times"?

multiply --- add exponents
raise to power - multiply exponents
a^.25 b^.5 * a^1 b^.5

a^1.25 b^1
or
a^(5/4) b

How do I write the answer in fraction form.

How do write this answer in fraction form?

To simplify the expression (a^(1/2) b)^(1/2) × (ab^(1/2)), we can use the product rule for exponents, as well as simplify any radicals involved.

Let's start by breaking down the expression step by step:

1. First, let's simplify the first radical term, (a^(1/2) b)^(1/2). According to the product rule for exponents, when we raise a product to a power, we can distribute that power to each factor within the parentheses.

So, (a^(1/2) b)^(1/2) can be simplified as:
(a^(1/2))^(1/2) × (b)^(1/2)
Now, when we raise a number to an exponent that is also an exponent (e.g., (a^(1/2))^(1/2)), we multiply the exponents together. In this case, we multiply 1/2 and 1/2 to get 1/4.

Therefore, (a^(1/2))^(1/2) × (b)^(1/2) simplifies to:
a^(1/4) × b^(1/2)

2. Now, let's simplify the second term, (ab^(1/2)). There are no radicals here, so we can leave it as it is.

Combining the simplified forms of the two terms, we have:
[a^(1/4) × b^(1/2)] × (ab^(1/2))

To multiply these terms together, we multiply the coefficients (the variables with the same base) and add the exponents of the variables with the same base.

Let's break down this expression further:

a^(1/4) × a × b^(1/2) × b^(1/2)
Simplifying, we have:
a^(1/4 + 1) × b^(1/2 + 1/2)
Which becomes:
a^(5/4) × b^(1)
Finally, we get the simplified form of the expression as:
a^(5/4) b

So, the simplified expression is a raised to the power of 5/4 multiplied by b.