If `x-(1)/(x)=7`, then what is the value of `(x^(3)-(1)/(x^(3)))`?
(a)350
(b)336
(c)343
(d)364

Options

A. 350
B. 336
C. 343
D. 364

Answer

To solve the equation \( x - \frac{1}{x} = 7 \) and find the value of \( x^3 - \frac{1}{x^3} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x - \frac{1}{x} = 7 \]

### Step 2: Cube both sides
To find \( x^3 - \frac{1}{x^3} \), we can use the identity:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
where \( a = x \) and \( b = \frac{1}{x} \). We can express \( x^3 - \frac{1}{x^3} \) in terms of \( x - \frac{1}{x} \):
\[ x^3 - \frac{1}{x^3} = \left( x - \frac{1}{x} \right) \left( x^2 + 1 + \frac{1}{x^2} \right) \]

### Step 3: Find \( x^2 + \frac{1}{x^2} \)
We can find \( x^2 + \frac{1}{x^2} \) using the square of \( x - \frac{1}{x} \):
\[ \left( x - \frac{1}{x} \right)^2 = x^2 - 2 + \frac{1}{x^2} \]
Substituting \( 7 \) into the equation:
\[ 7^2 = x^2 - 2 + \frac{1}{x^2} \]
\[ 49 = x^2 - 2 + \frac{1}{x^2} \]
Rearranging gives:
\[ x^2 + \frac{1}{x^2} = 49 + 2 = 51 \]

### Step 4: Substitute back to find \( x^3 - \frac{1}{x^3} \)
Now we can substitute \( x^2 + \frac{1}{x^2} \) back into our equation for \( x^3 - \frac{1}{x^3} \):
\[ x^3 - \frac{1}{x^3} = \left( x - \frac{1}{x} \right) \left( x^2 + 1 + \frac{1}{x^2} \right) \]
We already know:
- \( x - \frac{1}{x} = 7 \)
- \( x^2 + \frac{1}{x^2} = 51 \)

Thus:
\[ x^2 + 1 + \frac{1}{x^2} = 51 + 1 = 52 \]
Now substituting these values:
\[ x^3 - \frac{1}{x^3} = 7 \cdot 52 = 364 \]

### Final Answer
The value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{364} \]