If ` x^(4) + ((1)/( x^(4)))= 34` what is the value of `x^(3) - ((1)/( x^(3))) `?

Options

A. 0
B. 6
C. 8
D. 14

Answer

To solve the equation \( x^4 + \frac{1}{x^4} = 34 \) 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 equation:
\[ x^4 + \frac{1}{x^4} = 34 \]
We can rewrite this using the identity \( a^2 + b^2 = (a + b)^2 - 2ab \). Here, let \( a = x^2 \) and \( b = \frac{1}{x^2} \).

### Step 2: Add and subtract 2
To use the identity, we add and subtract 2:
\[ x^4 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2 \]
Thus, we can rewrite the equation as:
\[ (x^2 + \frac{1}{x^2})^2 - 2 = 34 \]

### Step 3: Solve for \( x^2 + \frac{1}{x^2} \)
Now, we can rearrange the equation:
\[ (x^2 + \frac{1}{x^2})^2 = 34 + 2 = 36 \]
Taking the square root of both sides gives:
\[ x^2 + \frac{1}{x^2} = 6 \]

### Step 4: Use the identity for cubes
Next, we need to find \( x^3 - \frac{1}{x^3} \). We can use the identity:
\[ x^3 - \frac{1}{x^3} = (x - \frac{1}{x}) \left( (x - \frac{1}{x})^2 + 3 \right) \]
First, we need to find \( x - \frac{1}{x} \).

### Step 5: Find \( x - \frac{1}{x} \)
We can use the identity:
\[ x^2 + \frac{1}{x^2} = (x - \frac{1}{x})^2 + 2 \]
Substituting \( x^2 + \frac{1}{x^2} = 6 \):
\[ 6 = (x - \frac{1}{x})^2 + 2 \]
Rearranging gives:
\[ (x - \frac{1}{x})^2 = 6 - 2 = 4 \]
Taking the square root:
\[ x - \frac{1}{x} = 2 \quad \text{or} \quad x - \frac{1}{x} = -2 \]

### Step 6: Substitute back to find \( x^3 - \frac{1}{x^3} \)
Using \( x - \frac{1}{x} = 2 \):
\[ x^3 - \frac{1}{x^3} = 2 \left( 4 + 3 \right) = 2 \times 7 = 14 \]
Using \( x - \frac{1}{x} = -2 \):
\[ x^3 - \frac{1}{x^3} = -2 \left( 4 + 3 \right) = -2 \times 7 = -14 \]

### Conclusion
Thus, the value of \( x^3 - \frac{1}{x^3} \) can be either \( 14 \) or \( -14 \), depending on the sign of \( x - \frac{1}{x} \).