If `x^(2) + (1)/(x^(2))= 7 and x ne 0`, find the value of : `7x^(3) + 8x- (7)/(x^(3))- (8)/(x)`

Answer

To solve the problem, we start with the given equation:

**Given:**
\[ x^2 + \frac{1}{x^2} = 7 \]

We need to find the value of:
\[ 7x^3 + 8x - \frac{7}{x^3} - \frac{8}{x} \]

### Step 1: Rearranging the Expression
We can rearrange the expression by grouping similar terms:
\[ 7x^3 - \frac{7}{x^3} + 8x - \frac{8}{x} \]
We can factor out 7 and 8:
\[ 7\left(x^3 - \frac{1}{x^3}\right) + 8\left(x - \frac{1}{x}\right) \]

### Step 2: Finding \( x^3 - \frac{1}{x^3} \)
To find \( x^3 - \frac{1}{x^3} \), we can use the identity:
\[ x^3 - \frac{1}{x^3} = \left(x - \frac{1}{x}\right)\left(x^2 + 1 + \frac{1}{x^2}\right) \]

First, we need to find \( x - \frac{1}{x} \).

### Step 3: Finding \( x - \frac{1}{x} \)
We know:
\[ x^2 + \frac{1}{x^2} = 7 \]
Using the identity:
\[ \left(x - \frac{1}{x}\right)^2 = x^2 - 2 + \frac{1}{x^2} \]
We can express it as:
\[ \left(x - \frac{1}{x}\right)^2 = 7 - 2 = 5 \]
Taking the square root gives us:
\[ x - \frac{1}{x} = \sqrt{5} \]

### Step 4: Finding \( x^2 + 1 + \frac{1}{x^2} \)
From the earlier expression:
\[ x^2 + \frac{1}{x^2} = 7 \]
Thus:
\[ x^2 + 1 + \frac{1}{x^2} = 7 + 1 = 8 \]

### Step 5: Substitute Back to Find \( x^3 - \frac{1}{x^3} \)
Now we can substitute back into the identity:
\[ x^3 - \frac{1}{x^3} = \left(x - \frac{1}{x}\right)\left(x^2 + 1 + \frac{1}{x^2}\right) \]
Substituting the values we found:
\[ x^3 - \frac{1}{x^3} = \sqrt{5} \cdot 8 = 8\sqrt{5} \]

### Step 6: Substitute Values into the Original Expression
Now we substitute \( x^3 - \frac{1}{x^3} \) and \( x - \frac{1}{x} \) back into the expression we rearranged:
\[ 7(x^3 - \frac{1}{x^3}) + 8(x - \frac{1}{x}) \]
This becomes:
\[ 7(8\sqrt{5}) + 8(\sqrt{5}) = 56\sqrt{5} + 8\sqrt{5} = 64\sqrt{5} \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{64\sqrt{5}} \]