The expression you provided is as follows:

$\left( \frac{9\sqrt{3} + 5\sqrt{11}}{6\sqrt{3}} \right)^{1/3} + \left( \frac{9\sqrt{3} - 5\sqrt{11}}{6\sqrt{3}} \right)^{1/3}$

Let's go step by step to simplify this.

Step 1: Simplify each of the terms inside the cube roots.

• First term:
  $\frac{9\sqrt{3} + 5\sqrt{11}}{6\sqrt{3}} = \frac{9\sqrt{3}}{6\sqrt{3}} + \frac{5\sqrt{11}}{6\sqrt{3}}$ Simplifying the first part: $\frac{9\sqrt{3}}{6\sqrt{3}} = \frac{9}{6} = \frac{3}{2}$ So, the first term becomes: $\frac{3}{2} + \frac{5\sqrt{11}}{6\sqrt{3}}$

• Second term:
  $\frac{9\sqrt{3} - 5\sqrt{11}}{6\sqrt{3}} = \frac{9\sqrt{3}}{6\sqrt{3}} - \frac{5\sqrt{11}}{6\sqrt{3}}$ Similar to the first term, the first part simplifies to: $\frac{9}{6} = \frac{3}{2}$ So, the second term becomes: $\frac{3}{2} - \frac{5\sqrt{11}}{6\sqrt{3}}$

Step 2: Analyze the structure of the expression.

Notice that the two terms inside the cube roots are very similar. They have the form $a + b$ and $a - b$, where:

• $a = \frac{3}{2}$
• $b = \frac{5\sqrt{11}}{6\sqrt{3}}$

This structure suggests we might be dealing with a sum of cubes or another identity. However, simplifying further requires detailed root evaluation, but the pattern here is clear that these terms are set up symmetrically, potentially leading to further simplification or cancellation.

Final Step: Conclusion

The final simplification would require advanced algebraic manipulation (potentially factoring the cube roots), but this structure is complex to simplify further without numeric approximation.

Would you like more details on how to proceed with numerical simplifications or explore any specific part of the process further?

Here are 5 related questions:

1. How do we rationalize the denominator when simplifying square roots?
2. What are the common properties of cube roots that might help simplify this expression?
3. How would you simplify if both terms under the cube root were exactly equal?
4. Can we factor cube root expressions using known identities like sum/difference of cubes?
5. What role do symmetry and patterns play in simplifying complex algebraic expressions?

Tip: When simplifying cube root expressions, look for patterns like $a + b$ and $a - b$, as they often suggest the use of identities or special factorizations.