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?

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 $a^{m/n} = (a^m)^{1/n} = \sqrt[n]{a^m}$.