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We are given the equation:

\[
x^2 + \frac{1}{x^2} = \frac{17}{4}
\]
where \( x > 0 \). We need to find the value of \( x^3 + \frac{1}{x^3} \).

### Step 1: Use identity for cubes
We know that:
\[
x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3 \left( x + \frac{1}{x} \right)
\]
First, we need to determine the value of \( x + \frac{1}{x} \) from the given equation.

### Step 2: Derive \( x + \frac{1}{x} \)
We have the identity:
\[
x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2
\]
Substitute the given value \( x^2 + \frac{1}{x^2} = \frac{17}{4} \) into this equation:
\[
\frac{17}{4} = \left( x + \frac{1}{x} \right)^2 - 2
\]
Adding 2 to both sides:
\[
\left( x + \frac{1}{x} \right)^2 = \frac{17}{4} + 2 = \frac{17}{4} + \frac{8}{4} = \frac{25}{4}
\]
Taking the square root of both sides:
\[
x + \frac{1}{x} = \frac{5}{2}
\]

### Step 3: Compute \( x^3 + \frac{1}{x^3} \)
Now, using the identity for cubes:
\[
x^3 + \frac{1}{x^3} = \left( \frac{5}{2} \right)^3 - 3 \times \frac{5}{2}
\]
First, calculate \( \left( \frac{5}{2} \right)^3 \):
\[
\left( \frac{5}{2} \right)^3 = \frac{125}{8}
\]
Now subtract \( 3 \times \frac{5}{2} \):
\[
x^3 + \frac{1}{x^3} = \frac{125}{8} - \frac{15}{2} = \frac{125}{8} - \frac{60}{8} = \frac{65}{8}
\]

Thus, the value of \( x^3 + \frac{1}{x^3} \) is \( \frac{65}{8} \).

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

Would you like further details on the solution? Here are five related questions you might explore:

1. How can you generalize the solution for different values of \( x^2 + \frac{1}{x^2} \)?
2. What are the key identities used in simplifying powers of binomials like \( x + \frac{1}{x} \)?
3. How do you solve similar problems when \( x^2 + \frac{1}{x^2} \) has complex values?
4. What other applications can the cube identities be used for in calculus or algebra?
5. How does this type of problem relate to solving polynomial equations?

**Tip**: Always remember to use identities like \( x^3 + \frac{1}{x^3} \) to simplify calculations without directly finding \( x \).

If x^2 + 1/x^2 = 17/4 and x > 0, find the value of x^3 + 1/x^3.

Learn how to solve the algebraic expression where x^2 + 1/x^2 equals 17/4, and find the value of x^3 + 1/x^3. This problem covers binomial identities and algebraic manipulation suitable for Grades 9-12.

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