We are given the equation \( x^4 + \frac{1}{x^4} = 119 \) and asked to find the value of \( x^3 - \frac{1}{x^3} \).

### Step 1: Express the given equation in terms of \( x^2 + \frac{1}{x^2} \)
We can manipulate the expression \( x^4 + \frac{1}{x^4} \) by recalling the identity:
\[
\left( x^2 + \frac{1}{x^2} \right)^2 = x^4 + \frac{1}{x^4} + 2
\]
So, we have:
\[
x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2
\]
Substitute \( x^4 + \frac{1}{x^4} = 119 \) into this equation:
\[
119 = \left( x^2 + \frac{1}{x^2} \right)^2 - 2
\]
Solving for \( \left( x^2 + \frac{1}{x^2} \right)^2 \):
\[
\left( x^2 + \frac{1}{x^2} \right)^2 = 119 + 2 = 121
\]
\[
x^2 + \frac{1}{x^2} = \sqrt{121} = 11
\]

### Step 2: Use \( x^2 + \frac{1}{x^2} \) to find \( x + \frac{1}{x} \)
Now, recall the identity:
\[
\left( x + \frac{1}{x} \right)^2 = x^2 + \frac{1}{x^2} + 2
\]
Substitute \( x^2 + \frac{1}{x^2} = 11 \) into this equation:
\[
\left( x + \frac{1}{x} \right)^2 = 11 + 2 = 13
\]
\[
x + \frac{1}{x} = \sqrt{13}
\]

### Step 3: Find \( x^3 - \frac{1}{x^3} \)
Now, to find \( x^3 - \frac{1}{x^3} \), use the identity:
\[
\left( x + \frac{1}{x} \right)^3 = x^3 + \frac{1}{x^3} + 3 \left( x + \frac{1}{x} \right)
\]
Substitute \( x + \frac{1}{x} = \sqrt{13} \) into this equation:
\[
\left( \sqrt{13} \right)^3 = x^3 + \frac{1}{x^3} + 3 \sqrt{13}
\]
Simplifying:
\[
\sqrt{13}^3 = 13 \sqrt{13}
\]
Thus, the equation becomes:
\[
13 \sqrt{13} = x^3 + \frac{1}{x^3} + 3 \sqrt{13}
\]
Solving for \( x^3 - \frac{1}{x^3} \):
\[
x^3 - \frac{1}{x^3} = \boxed{-3 \sqrt{13}}
\]

Do you have any questions or would you like further details?

Here are 5 related questions:
1. How would you solve for \( x^5 - \frac{1}{x^5} \) given similar conditions?
2. What is the general method for finding higher powers of \( x + \frac{1}{x} \)?
3. Can you solve for \( x + \frac{1}{x} \) directly from \( x^4 + \frac{1}{x^4} \)?
4. What if the original equation was \( x^4 - \frac{1}{x^4} = 119 \)?
5. How can you apply these identities to complex numbers?

**Tip:** When solving equations involving powers of sums and reciprocals, always check for key algebraic identities to simplify the process.

Learn how to solve the equation x⁴ + 1/x⁴ = 119 and find the value of x³ - 1/x³ with a step-by-step solution. This problem involves algebraic identities and power manipulation, suitable for students in Grades 9-12.

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