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The question asks us to find the value of \( x^3 + \frac{1}{x^3} \) given that:

\[
x^2 + \frac{1}{x^2} = \frac{17}{4}, \quad x > 0.
\]

We can solve this using the following steps:

### Step 1: Start with the given equation
\[
x^2 + \frac{1}{x^2} = \frac{17}{4}
\]

### Step 2: Find \( x^3 + \frac{1}{x^3} \)
To find \( x^3 + \frac{1}{x^3} \), we use the identity:

\[
\left( x + \frac{1}{x} \right)^2 = x^2 + \frac{1}{x^2} + 2
\]

Substitute the given value \( x^2 + \frac{1}{x^2} = \frac{17}{4} \):

\[
\left( x + \frac{1}{x} \right)^2 = \frac{17}{4} + 2
\]

Convert 2 to a fraction with denominator 4:

\[
\left( x + \frac{1}{x} \right)^2 = \frac{17}{4} + \frac{8}{4} = \frac{25}{4}
\]

Taking the square root of both sides:

\[
x + \frac{1}{x} = \frac{5}{2}
\]

### Step 3: Use the identity for \( x^3 + \frac{1}{x^3} \)
We now use the identity:

\[
x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3\left( x + \frac{1}{x} \right)
\]

Substitute \( x + \frac{1}{x} = \frac{5}{2} \):

\[
x^3 + \frac{1}{x^3} = \left( \frac{5}{2} \right)^3 - 3\left( \frac{5}{2} \right)
\]

Simplify the terms:

\[
\left( \frac{5}{2} \right)^3 = \frac{125}{8}
\]
\[
3 \times \frac{5}{2} = \frac{15}{2} = \frac{60}{8}
\]

Thus:

\[
x^3 + \frac{1}{x^3} = \frac{125}{8} - \frac{60}{8} = \frac{65}{8}
\]

### Final Answer:
The value of \( x^3 + \frac{1}{x^3} \) is \( \frac{65}{8} \).

The correct answer is option (C) \( \frac{65}{8} \).

---

Would you like further clarification or details? Here are five related questions to deepen your understanding:

1. How can we derive other identities like \( x^4 + \frac{1}{x^4} \) using similar methods?
2. What other forms of algebraic identities are useful for solving equations involving higher powers of \( x \)?
3. Can this approach be applied to complex numbers, and how would the results change?
4. How would the solution change if \( x^2 + \frac{1}{x^2} \) had a different value, say \( \frac{25}{9} \)?
5. How can we interpret these algebraic manipulations geometrically?

**Tip**: When working with algebraic identities, memorizing common patterns (like those for cubes and squares) can help you solve complex expressions faster.

If x^2 + 1/x^2 = 17/4, x > 0, then what is the value of x^3 + 1/x^3?

Learn how to solve an algebraic expression involving x^3 + 1/x^3 given x^2 + 1/x^2 = 17/4. This problem uses key algebraic identities and is suitable for high school students in Grades 10-12.

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