The image shows two expressions involving cube roots and square roots. Let's break them down for clarity:

1. Expression $a)$: [ P = \sqrt[3]{26 + 15\sqrt{3}} - \sqrt[3]{6\sqrt{3} - 10} ]

2. Expression $b)$: [ Q = \sqrt[3]{17\sqrt{5} - 38} + 0 + \sqrt[3]{72 - 32\sqrt{5}} ]

Now, let's solve them step by step.

a) $P = \sqrt[3]{26 + 15\sqrt{3}} - \sqrt[3]{6\sqrt{3} - 10}$

We can evaluate each cube root individually to approximate the value of $P$.

b) $Q = \sqrt[3]{17\sqrt{5} - 38} + \sqrt[3]{72 - 32\sqrt{5}}$

Again, each cube root term can be calculated separately and added together to find $Q$.

I'll compute the approximate values for both expressions. Let's proceed with that.The approximate values for the expressions are:

• $P \approx 3$
• $Q \approx 1$

Would you like more details or have any further questions? Here are some related questions that might help deepen your understanding:

1. How are cube roots calculated for expressions involving radicals?
2. Can cube roots always be simplified to exact values like integers?
3. What strategies can be used for simplifying expressions involving multiple radicals?
4. How does the evaluation of nested square and cube roots differ from simpler radical operations?
5. Are there general rules for solving cube roots of sums involving square roots?

Tip: When solving cube roots of complex expressions, approximate numerical values can often help guide intuition before trying for exact algebraic simplifications.