If ` x - (1)/(x) = y , x ne 0 , `find the value of ` (x- (1)/ (x) - 2y ) ^(3)`

Answer

To solve the problem step by step, we start with the given equation and the expression we need to evaluate.

### Step 1: Write down the given equation
We have:
\[ x - \frac{1}{x} = y \]
where \( x \neq 0 \).

### Step 2: Substitute \( y \) in the expression
We need to find the value of:
\[ \left( x - \frac{1}{x} - 2y \right)^3 \]
We can substitute \( y \) from the first equation into this expression.

### Step 3: Substitute \( y \) into the expression
Substituting \( y \) gives us:
\[ x - \frac{1}{x} - 2y = x - \frac{1}{x} - 2\left(x - \frac{1}{x}\right) \]
This simplifies to:
\[ x - \frac{1}{x} - 2x + \frac{2}{x} \]

### Step 4: Combine like terms
Now, combine the terms:
\[ (x - 2x) + \left(-\frac{1}{x} + \frac{2}{x}\right) = -x + \frac{1}{x} \]
Thus, we have:
\[ -\left(x - \frac{1}{x}\right) \]

### Step 5: Substitute back for \( y \)
Since \( x - \frac{1}{x} = y \), we can substitute back:
\[ -\left(x - \frac{1}{x}\right) = -y \]

### Step 6: Cube the expression
Now we need to cube this result:
\[ \left(-y\right)^3 = -y^3 \]

### Final Result
Thus, the value of the expression \( \left( x - \frac{1}{x} - 2y \right)^3 \) is:
\[ -y^3 \]

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