https://file.aiquickdraw.com/mathbot/file/1725878397434a6tbls8r.jpg

The expression in the image is as follows:

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

Let's break this down step by step:

1. **First Term**: \(\left(\frac{a^{\frac{2}{3}}}{b^{\frac{1}{2}}}\right)^{-1}\)
   - Applying the negative exponent rule: \(\left(\frac{a^{\frac{2}{3}}}{b^{\frac{1}{2}}}\right)^{-1} = \frac{b^{\frac{1}{2}}}{a^{\frac{2}{3}}}\)

2. **Second Term**: \(\left(a^{\frac{2}{3}} b^{\frac{1}{3}}\right)^2\)
   - Apply the exponent rule: \(\left(a^{\frac{2}{3}} b^{\frac{1}{3}}\right)^2 = a^{\frac{4}{3}} b^{\frac{2}{3}}\)

3. **Third Term**: \(\frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}}\)

Now, combining everything:

\[
\frac{b^{\frac{1}{2}}}{a^{\frac{2}{3}}} \times a^{\frac{4}{3}} b^{\frac{2}{3}} \div \frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}}
\]

Next, simplify step by step:

4. **Combine the first and second terms**:
   \[
   \frac{b^{\frac{1}{2}} \times a^{\frac{4}{3}} \times b^{\frac{2}{3}}}{a^{\frac{2}{3}}} = \frac{a^{\frac{4}{3}} \times b^{\frac{1}{2} + \frac{2}{3}}}{a^{\frac{2}{3}}} = \frac{a^{\frac{4}{3}} \times b^{\frac{7}{6}}}{a^{\frac{2}{3}}}
   \]
   Simplifying further:
   \[
   a^{\frac{4}{3} - \frac{2}{3}} \times b^{\frac{7}{6}} = a^{\frac{2}{3}} \times b^{\frac{7}{6}}
   \]

5. **Now, divide by the third term**:
   \[
   \frac{a^{\frac{2}{3}} \times b^{\frac{7}{6}}}{\frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}}} = a^{\frac{2}{3}} \times b^{\frac{7}{6}} \times \frac{a^{\frac{1}{3}}}{b^{\frac{1}{2}}}
   \]

6. **Simplify the final expression**:
   \[
   a^{\frac{2}{3} + \frac{1}{3}} \times b^{\frac{7}{6} - \frac{1}{2}} = a^1 \times b

Simplify the expression (a^(2/3) / b^(1/2))^(-1) * (a^(2/3) * b^(1/3))^2 ÷ (b^(1/2) / a^(1/3))

Learn how to simplify the expression (a^(2/3) / b^(1/2))^(-1) * (a^(2/3) * b^(1/3))^2 ÷ (b^(1/2) / a^(1/3)). This problem focuses on fractional exponents, exponent rules, and simplifying complex algebraic expressions. Suitable for high school students.

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