Let's solve the given expression step-by-step.

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The expression contains:

1. First fraction: [ \frac{\sqrt[3]{(4 + \sqrt{17})^2} + \sqrt{17}}{\sqrt[3]{4 - \sqrt{17}}} ]

2. Second product: [ \sqrt[4]{9 - \sqrt{65}} \cdot \sqrt[4]{9 + \sqrt{65}} ]

3. Third product: $\sqrt{3 - \sqrt{5}} \cdot \sqrt{3 + \sqrt{5}}$

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Step 1: Simplify the first fraction.

Numerator: $\sqrt[3]{(4 + \sqrt{17})^2} + \sqrt{17}$

• We leave the cube root and $+ \sqrt{17}$ in their original forms for now.

Denominator: $\sqrt[3]{4 - \sqrt{17}}$

• This part remains unchanged.

Thus, the first term simplifies as a fraction in terms of cube roots.

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Step 2: Simplify the second product.

Using the property $(\sqrt[a]{x} \cdot \sqrt[a]{y} = \sqrt[a]{xy})$, we combine the terms:

$\sqrt[4]{(9 - \sqrt{65})(9 + \sqrt{65})}$

Now simplify the product inside the root: $(9 - \sqrt{65})(9 + \sqrt{65}) = 9^2 - (\sqrt{65})^2 = 81 - 65 = 16.$

Thus: $\sqrt[4]{16} = 2.$

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Step 3: Simplify the third product.

Similarly, combine terms under a square root: $\sqrt{(3 - \sqrt{5})(3 + \sqrt{5})}.$

Simplify the product inside the root: $(3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4.$

Thus: $\sqrt{4} = 2.$

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Final Answer:

1. The first fraction remains unevaluated in its current form: [ \frac{\sqrt[3]{(4 + \sqrt{17})^2} + \sqrt{17}}{\sqrt[3]{4 - \sqrt{17}}} ]

2. The second product simplifies to: $2$

3. The third product simplifies to: $2$

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Let me know if you'd like further details about the steps or have specific questions! Here are five related questions:

1. How can cube roots be rationalized or simplified?
2. What are other properties of radicals for simplifying products?
3. Can this expression be simplified further with approximations?
4. How do nested radicals like $(4 + \sqrt{17})$ behave in equations?
5. How are fourth roots commonly used in algebra?

Tip: Always check if the radicals can simplify by using properties like $(a + b)(a - b)$. This often helps in evaluating products quickly.