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The image shows a mathematical expression involving exponents and fractions. The expression is as follows:

\[
\left( \frac{a^{\frac{2}{3}}}{b^{\frac{1}{2}}} \right)^{-1} \cdot \left( a^{\frac{2}{3}} b^{\frac{1}{3}} \right)^2 : \left( \frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}} \right)
\]

Let me solve this step by step.

### Step 1: Inverse the first term
The first term is raised to the power of \(-1\), which means we take the reciprocal of the term:
\[
\left( \frac{a^{\frac{2}{3}}}{b^{\frac{1}{2}}} \right)^{-1} = \frac{b^{\frac{1}{2}}}{a^{\frac{2}{3}}}
\]

### Step 2: Square the second term
The second term is \(\left( a^{\frac{2}{3}} b^{\frac{1}{3}} \right)^2\). Apply the exponent to both factors:
\[
\left( a^{\frac{2}{3}} b^{\frac{1}{3}} \right)^2 = a^{\frac{4}{3}} b^{\frac{2}{3}}
\]

### Step 3: Simplify the third term
The third term is \(\left( \frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}} \right)\), so this remains as is for now.

### Step 4: Combine everything
Now substitute everything back into the expression:
\[
\frac{b^{\frac{1}{2}}}{a^{\frac{2}{3}}} \cdot a^{\frac{4}{3}} b^{\frac{2}{3}} : \frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}}
\]

The colon (\(:\)) represents division. So, rewrite it as multiplication by the reciprocal:
\[
= \frac{b^{\frac{1}{2}}}{a^{\frac{2}{3}}} \cdot a^{\frac{4}{3}} b^{\frac{2}{3}} \cdot \frac{a^{\frac{1}{3}}}{b^{\frac{1}{2}}}
\]

### Step 5: Simplify powers of \(a\) and \(b\)
Now, simplify powers of \(a\) and \(b\):

For \(a\):
\[
a^{-\frac{2}{3}} \cdot a^{\frac{4}{3}} \cdot a^{\frac{1}{3}} = a^{-\frac{2}{3} + \frac{4}{3} + \frac{1}{3}} = a^{\frac{3}{3}} = a
\]

For \(b\):
\[
b^{\frac{1}{2}} \cdot b^{\frac{2}{3}} \cdot b^{-\frac{1}{2}} = b^{\frac{1}{2} + \frac{2}{3} - \frac{1}{2}} = b^{\frac{2}{3}}
\]

### Final Simplified Expression:
\[
a b^{\frac{2}{3}}
\]

So, the simplified result is:
\[
a b^{\frac{2}{3}}
\]

Would you like further clarification on any steps or details? 

Here are some related questions for further exploration:
1. How do you simplify expressions with negative exponents?
2. What are the rules for multiplying terms with fractional exponents?
3. How do you handle division with powers in the same base?
4. Can fractional exponents be converted into radicals?
5. How would this change if the exponents involved negative or complex numbers?

**Tip:** When simplifying expressions involving exponents, always look for opportunities to combine terms with the same base to simplify calculations.

Simplify the given expression involving exponents and fractions.

Explore how to simplify complex expressions with exponents and fractions. This guide covers reciprocal of exponentiation, power of products, and simplifying expressions using laws of exponents, ideal for high school students.

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