If `x ^(4) +(1)/(x ^(4))= 119,` then the value of `x ^(3) - (1)/(x ^(3))` is :

Options

A. `-36`
B. `pm 36`
C. `pm 33`
D. `36`

Answer

To solve the equation \( x^4 + \frac{1}{x^4} = 119 \) and find the value of \( x^3 - \frac{1}{x^3} \), we can follow these steps:

### Step 1: Define \( k \)
Let \( k = x^4 + \frac{1}{x^4} = 119 \).

### Step 2: Find \( x^2 + \frac{1}{x^2} \)
Using the identity:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
We can rearrange it to find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = \sqrt{k + 2} \]
Substituting \( k = 119 \):
\[ x^2 + \frac{1}{x^2} = \sqrt{119 + 2} = \sqrt{121} = 11 \]

### Step 3: Find \( x + \frac{1}{x} \)
Using the identity:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
We can rearrange it to find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = \sqrt{x^2 + \frac{1}{x^2} + 2} = \sqrt{11 + 2} = \sqrt{13} \]

### Step 4: Find \( x^3 - \frac{1}{x^3} \)
Using the identity:
\[ x^3 - \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) \]
Substituting the values we have:
\[ x^3 - \frac{1}{x^3} = \sqrt{13} \cdot 11 - \sqrt{13} = 11\sqrt{13} - \sqrt{13} = 10\sqrt{13} \]

### Step 5: Find \( x^3 - \frac{1}{x^3} \) in terms of numerical values
To find the numerical value of \( x^3 - \frac{1}{x^3} \), we can evaluate \( 10\sqrt{13} \):
\[ 10\sqrt{13} \approx 10 \times 3.605 = 36.05 \]
Thus, the value of \( x^3 - \frac{1}{x^3} \) is approximately \( 36 \).

### Final Answer
The value of \( x^3 - \frac{1}{x^3} \) is \( \pm 36 \).