If `x^4 + (1)/(x^4) =727` , then find the value of `x^3 - (1)/(x^3)`

Options

A. 140
B. 120
C. 190
D. 160

Answer

To solve the equation \( x^4 + \frac{1}{x^4} = 727 \) and find the value of \( x^3 - \frac{1}{x^3} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x^4 + \frac{1}{x^4} = 727 \]

### Step 2: Add 2 to both sides
We add 2 to both sides to prepare for a square:
\[ x^4 + \frac{1}{x^4} + 2 = 727 + 2 \]
This simplifies to:
\[ x^4 + \frac{1}{x^4} + 2 = 729 \]

### Step 3: Recognize the square
The left-hand side can be factored as:
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 729 \]

### Step 4: Take the square root
Now we take the square root of both sides:
\[ x^2 + \frac{1}{x^2} = \sqrt{729} \]
Since \( \sqrt{729} = 27 \), we have:
\[ x^2 + \frac{1}{x^2} = 27 \]

### Step 5: Subtract 2 from both sides
Next, we subtract 2 from both sides to find \( x - \frac{1}{x} \):
\[ x^2 + \frac{1}{x^2} - 2 = 27 - 2 \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} - 2 = 25 \]

### Step 6: Recognize the square
The left-hand side can also be factored:
\[ \left( x - \frac{1}{x} \right)^2 = 25 \]

### Step 7: Take the square root
Taking the square root gives us:
\[ x - \frac{1}{x} = \sqrt{25} = 5 \]
Thus, we have:
\[ x - \frac{1}{x} = 5 \]

### Step 8: Find \( x^3 - \frac{1}{x^3} \)
Now we can 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 know \( x - \frac{1}{x} = 5 \). Now we need to find \( x^2 + 1 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 27 \implies x^2 + 1 + \frac{1}{x^2} = 27 + 1 = 28 \]

### Step 9: Substitute values
Now we substitute back into the identity:
\[ x^3 - \frac{1}{x^3} = 5 \cdot 28 = 140 \]

### Final Answer
Thus, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{140} \]