If `x+ (1)/(x)=3`, then
`x^(5) + (1)/(x^(5))`= ?

Answer

To solve the equation \( x + \frac{1}{x} = 3 \) and find \( x^5 + \frac{1}{x^5} \), we can follow these steps:

### Step 1: Define \( y = x + \frac{1}{x} \)
We start by defining \( y \) as:
\[ y = x + \frac{1}{x} \]
From the problem, we know that:
\[ y = 3 \]

### Step 2: Find \( x^2 + \frac{1}{x^2} \)
We can derive \( x^2 + \frac{1}{x^2} \) using the identity:
\[ x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 \]
Substituting \( y \) into the equation:
\[ x^2 + \frac{1}{x^2} = y^2 - 2 = 3^2 - 2 = 9 - 2 = 7 \]

### Step 3: Find \( x^3 + \frac{1}{x^3} \)
Next, we can find \( x^3 + \frac{1}{x^3} \) using the identity:
\[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})(x^2 + \frac{1}{x^2}) - (x + \frac{1}{x}) \]
Substituting the values we have:
\[ x^3 + \frac{1}{x^3} = y \cdot (x^2 + \frac{1}{x^2}) - y = 3 \cdot 7 - 3 = 21 - 3 = 18 \]

### Step 4: Find \( x^4 + \frac{1}{x^4} \)
Now, we can find \( x^4 + \frac{1}{x^4} \) using:
\[ x^4 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2 \]
Substituting the value we found:
\[ x^4 + \frac{1}{x^4} = 7^2 - 2 = 49 - 2 = 47 \]

### Step 5: Find \( x^5 + \frac{1}{x^5} \)
Finally, we can find \( x^5 + \frac{1}{x^5} \) using:
\[ x^5 + \frac{1}{x^5} = (x + \frac{1}{x})(x^4 + \frac{1}{x^4}) - (x^3 + \frac{1}{x^3}) \]
Substituting the values:
\[ x^5 + \frac{1}{x^5} = y \cdot (x^4 + \frac{1}{x^4}) - (x^3 + \frac{1}{x^3}) = 3 \cdot 47 - 18 = 141 - 18 = 123 \]

### Final Answer
Thus, the value of \( x^5 + \frac{1}{x^5} \) is:
\[ \boxed{123} \]