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

Options

A. 49
B. 27
C. 36
D. 63

Answer

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

### Step 1: Rewrite the equation
We start with the equation:
\[ x^4 + \frac{1}{x^4} = 119 \]
We can add 2 to both sides:
\[ x^4 + \frac{1}{x^4} + 2 = 119 + 2 \]
This simplifies to:
\[ x^4 + \frac{1}{x^4} + 2 = 121 \]

### Step 2: Recognize the square
Notice that \( x^4 + \frac{1}{x^4} + 2 \) can be rewritten using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \). Here, let \( a = x^2 \) and \( b = \frac{1}{x^2} \):
\[ (x^2 + \frac{1}{x^2})^2 = 121 \]

### Step 3: Take the square root
Taking the square root of both sides gives:
\[ x^2 + \frac{1}{x^2} = \sqrt{121} = 11 \quad \text{(we only take the positive root since squares are non-negative)} \]

### Step 4: Find \( x - \frac{1}{x} \)
We can now find \( x - \frac{1}{x} \) using the identity:
\[ x^2 + \frac{1}{x^2} = (x - \frac{1}{x})^2 + 2 \]
Substituting the value we found:
\[ 11 = (x - \frac{1}{x})^2 + 2 \]
Subtracting 2 from both sides:
\[ (x - \frac{1}{x})^2 = 9 \]
Taking the square root:
\[ x - \frac{1}{x} = \sqrt{9} = 3 \quad \text{(again, we take the positive root)} \]

### Step 5: Find \( x^3 - \frac{1}{x^3} \)
Now we can find \( x^3 - \frac{1}{x^3} \) using the identity:
\[ x^3 - \frac{1}{x^3} = (x - \frac{1}{x}) \left((x - \frac{1}{x})^2 + 3\right) \]
Substituting the value we found:
\[ x^3 - \frac{1}{x^3} = 3 \left(3^2 + 3\right) \]
Calculating:
\[ = 3 \left(9 + 3\right) = 3 \times 12 = 36 \]

### Final Answer
Thus, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{36} \]