The problem in the image asks for the value of the expression:

$\left( \frac{a^3}{b^4} \right)^{\frac{1}{2}} \times \left( a^{\frac{1}{2}} / b \right)$

Let's simplify this step by step.

Step 1: Simplify the first part of the expression

$\left( \frac{a^3}{b^4} \right)^{\frac{1}{2}} = \frac{(a^3)^{\frac{1}{2}}}{(b^4)^{\frac{1}{2}}} = \frac{a^{\frac{3}{2}}}{b^2}$

Step 2: Simplify the second part of the expression

$\frac{a^{\frac{1}{2}}}{b}$

Step 3: Multiply the simplified expressions

Now, multiply the two simplified expressions together: $\frac{a^{\frac{3}{2}}}{b^2} \times \frac{a^{\frac{1}{2}}}{b} = \frac{a^{\frac{3}{2}} \times a^{\frac{1}{2}}}{b^2 \times b}$ $= \frac{a^{\frac{3}{2} + \frac{1}{2}}}{b^{2+1}} = \frac{a^2}{b^3}$

This matches option (E) in the provided answers.

Final Answer:

$\boxed{ab}$

Do you have any questions or need further details?

Here are five questions related to the problem:

1. How do you simplify expressions with exponents?
2. What are the rules for multiplying terms with exponents?
3. How do you handle fractional exponents in an expression?
4. What is the significance of simplifying expressions before multiplying?
5. How do you multiply fractions involving algebraic terms?

Tip: When dealing with exponents, always remember that you add the exponents when multiplying terms with the same base.