If `x=(7-sqrt(45))/(2)`, find the value of `(x^(3)+(1)/(x^(3)))-7(x^(2)+(1)/(x^(2)))+(x+(1)/(x))` .

Answer

To solve the problem, we need to find the value of the expression \( (x^3 + \frac{1}{x^3}) - 7(x^2 + \frac{1}{x^2}) + (x + \frac{1}{x}) \) given that \( x = \frac{7 - \sqrt{45}}{2} \).

### Step 1: Find \( x + \frac{1}{x} \)

First, we need to calculate \( x + \frac{1}{x} \).

1. **Calculate \( \frac{1}{x} \)**:
\[ \frac{1}{x} = \frac{2}{7 - \sqrt{45}} \]
To rationalize this, we multiply the numerator and denominator by \( 7 + \sqrt{45} \):
\[ \frac{1}{x} = \frac{2(7 + \sqrt{45})}{(7 - \sqrt{45})(7 + \sqrt{45})} = \frac{2(7 + \sqrt{45})}{49 - 45} = \frac{2(7 + \sqrt{45})}{4} = \frac{7 + \sqrt{45}}{2} \]

2. **Now calculate \( x + \frac{1}{x} \)**:
\[ x + \frac{1}{x} = \frac{7 - \sqrt{45}}{2} + \frac{7 + \sqrt{45}}{2} = \frac{(7 - \sqrt{45}) + (7 + \sqrt{45})}{2} = \frac{14}{2} = 7 \]

### Step 2: Find \( x^2 + \frac{1}{x^2} \)

Using the identity:
\[ x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 \]
Substituting the value we found:
\[ x^2 + \frac{1}{x^2} = 7^2 - 2 = 49 - 2 = 47 \]

### Step 3: Find \( x^3 + \frac{1}{x^3} \)

Using the identity:
\[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 - 3(x + \frac{1}{x}) \]
Substituting the value we found:
\[ x^3 + \frac{1}{x^3} = 7^3 - 3 \cdot 7 = 343 - 21 = 322 \]

### Step 4: Substitute into the original expression

Now we substitute \( x^3 + \frac{1}{x^3} \), \( x^2 + \frac{1}{x^2} \), and \( x + \frac{1}{x} \) into the expression:
\[ (x^3 + \frac{1}{x^3}) - 7(x^2 + \frac{1}{x^2}) + (x + \frac{1}{x}) \]
Substituting the values:
\[ = 322 - 7 \cdot 47 + 7 \]
Calculating \( 7 \cdot 47 \):
\[ 7 \cdot 47 = 329 \]
Now substituting back:
\[ = 322 - 329 + 7 = 322 - 329 + 7 = 0 \]

### Final Answer
The value of the expression is \( 0 \).