https://file.aiquickdraw.com/mathbot/file/1730405906204vko0vvws.jpg

To solve the expression shown in the image, we’ll go through it step by step. The expression can be interpreted as follows:

\[
\frac{\left(9^{\frac{2}{3}} \times 6^{\frac{5}{3}}\right)}{2^{\frac{1}{3}} \sqrt[3]{9} \times 3^{\frac{1}{3}} \sqrt[3]{2}}
\]

Let’s simplify each part individually.

### Step 1: Simplify the Numerator

The numerator is:
\[
9^{\frac{2}{3}} \times 6^{\frac{5}{3}}
\]

1. **Calculate \( 9^{\frac{2}{3}} \):**
   - \( 9 = 3^2 \), so \( 9^{\frac{2}{3}} = (3^2)^{\frac{2}{3}} = 3^{\frac{4}{3}} \).

2. **Calculate \( 6^{\frac{5}{3}} \):**
   - \( 6 = 2 \times 3 \), so \( 6^{\frac{5}{3}} = (2 \times 3)^{\frac{5}{3}} = 2^{\frac{5}{3}} \times 3^{\frac{5}{3}} \).

Now, combining these results in the numerator:
\[
9^{\frac{2}{3}} \times 6^{\frac{5}{3}} = 3^{\frac{4}{3}} \times 2^{\frac{5}{3}} \times 3^{\frac{5}{3}} = 2^{\frac{5}{3}} \times 3^{\frac{9}{3}} = 2^{\frac{5}{3}} \times 3^3
\]

Since \( 3^3 = 27 \), the simplified form of the numerator is:
\[
27 \times 2^{\frac{5}{3}}
\]

### Step 2: Simplify the Denominator

The denominator is:
\[
2^{\frac{1}{3}} \sqrt[3]{9} \times 3^{\frac{1}{3}} \sqrt[3]{2}
\]

1. **Simplify \( \sqrt[3]{9} \):**
   - \( 9 = 3^2 \), so \( \sqrt[3]{9} = 3^{\frac{2}{3}} \).

2. **Simplify \( \sqrt[3]{2} \):**
   - This is just \( 2^{\frac{1}{3}} \).

Now rewrite the denominator using these simplifications:
\[
2^{\frac{1}{3}} \times 3^{\frac{2}{3}} \times 3^{\frac{1}{3}} \times 2^{\frac{1}{3}} = 2^{\frac{2}{3}} \times 3^{\frac{3}{3}} = 2^{\frac{2}{3}} \times 3
\]

### Step 3: Divide the Numerator by the Denominator

Now we have:
\[
\frac{27 \times 2^{\frac{5}{3}}}{2^{\frac{2}{3}} \times 3}
\]

Separate the constants and simplify the powers of 2:
\[
= \frac{27}{3} \times \frac{2^{\frac{5}{3}}}{2^{\frac{2}{3}}} = 9 \times 2^{\frac{3}{3}} = 9 \times 2 = 18
\]

### Final Answer

The simplified result is:
\[
18
\]

Would you like further details, or do you have any questions?

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Here are 5 related questions for practice:
1. How would you simplify \(\frac{8^{\frac{4}{3}}}{2^{\frac{4}{3}}}\)?
2. What is the result of \(9^{\frac{2}{3}} \times 4^{\frac{1}{3}}\)?
3. How would you simplify \(\frac{27^{\frac{2}{3}}}{3^{\frac{5}{3}}}\)?
4. What is \(\sqrt[3]{64} \times \sqrt[3]{16}\)?
5. How can you express \(6^{\frac{7}{3}}\) in terms of powers of 2 and 3?

**Tip:** When working with fractional exponents, remember that \(a^{m/n} = (a^m)^{1/n} = \sqrt[n]{a^m}\).

Simplify the expression \( \frac{ \left(9^{\frac{2}{3}} \times 6^{\frac{5}{3}}\right) }{ 2^{\frac{1}{3}} \sqrt[3]{9} \times 3^{\frac{1}{3}} \sqrt[3]{2} } \)

Learn how to simplify expressions involving fractional exponents and cube roots with a detailed step-by-step solution. This example uses properties of exponents to simplify complex expressions, suitable for students in Grades 10-12.

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