The problem you uploaded asks to simplify the following expression:

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

Step 1: Simplify $\left(\frac{a^{\frac{2}{3}}}{b^{\frac{1}{2}}}\right)^{-1}$

When you take the inverse (raise to the power of $-1$), you flip the fraction:

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

Step 2: Simplify the second part $\frac{(a^{\frac{2}{3}} b^{\frac{1}{2}})^2 \times a^{\frac{1}{3}}}{b^{\frac{1}{2}}}$

First, apply the power of 2 to both $a^{\frac{2}{3}}$ and $b^{\frac{1}{2}}$:

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

Now, multiply by $a^{\frac{1}{3}}$:

$a^{\frac{4}{3}} \times a^{\frac{1}{3}} = a^{\frac{5}{3}}$

So the expression becomes:

$\frac{a^{\frac{5}{3}} b^1}{b^{\frac{1}{2}}} = a^{\frac{5}{3}} b^{\frac{1}{2}}$

Step 3: Combine both parts

Now, multiply the two simplified expressions:

$\frac{b^{\frac{1}{2}}}{a^{\frac{2}{3}}} \times a^{\frac{5}{3}} b^{\frac{1}{2}} = \frac{a^{\frac{5}{3}}}{a^{\frac{2}{3}}} \times \frac{b^{\frac{1}{2}}}{1} = a^1 b^1 = ab$

Final Answer: $\boxed{ab}$

This matches option d.

Would you like more details or clarifications on this solution?

Here are 5 related questions for further practice:

1. How do you simplify expressions involving fractional exponents?
2. What happens when you multiply powers of the same base?
3. How does taking the inverse affect the exponents in a fraction?
4. What are the rules for dividing expressions with exponents?
5. How would the result change if the exponents were negative?

Tip: Always break down complex expressions step by step, focusing on simplifying one part at a time!