If `x^(4) +(1)/(x^(4)) =119 and x gt 1` then what is the value of `x^(3)-(1)/(x^(3))` ?

Options

A. 54
B. 18
C. 72
D. 36

Answer

To solve the problem, we need to find the value of \( x^3 - \frac{1}{x^3} \) given that \( x^4 + \frac{1}{x^4} = 119 \) and \( x > 1 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x^4 + \frac{1}{x^4} = 119 \]

2. **Use the identity**:
We know that:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
Thus, we can rewrite the equation as:
\[ \left( x^2 + \frac{1}{x^2} \right)^2 - 2 = 119 \]

3. **Add 2 to both sides**:
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 119 + 2 \]
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 121 \]

4. **Take the square root of both sides**:
\[ x^2 + \frac{1}{x^2} = \sqrt{121} = 11 \]
Since \( x > 1 \), we take the positive root.

5. **Use the identity again**:
We know that:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Therefore, we can write:
\[ \left( x + \frac{1}{x} \right)^2 - 2 = 11 \]

6. **Add 2 to both sides**:
\[ \left( x + \frac{1}{x} \right)^2 = 11 + 2 \]
\[ \left( x + \frac{1}{x} \right)^2 = 13 \]

7. **Take the square root of both sides**:
\[ x + \frac{1}{x} = \sqrt{13} \]
Again, since \( x > 1 \), we take the positive root.

8. **Now find \( x^3 - \frac{1}{x^3} \)**:
We use the identity:
\[ x^3 - \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 - 1 + \frac{1}{x^2} \right) \]
We already have \( x + \frac{1}{x} = \sqrt{13} \) and we need \( x^2 - 1 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 11 \implies x^2 - 1 + \frac{1}{x^2} = 11 - 1 = 10 \]

9. **Substituting back**:
\[ x^3 - \frac{1}{x^3} = \sqrt{13} \cdot 10 \]
\[ x^3 - \frac{1}{x^3} = 10\sqrt{13} \]

### Final Value:
Thus, the value of \( x^3 - \frac{1}{x^3} \) is \( 10\sqrt{13} \).