If `x+ (1)/(x)=3`, then
`x^(6) + (1)/(x^(6))=`?

Answer

To solve the equation \( x + \frac{1}{x} = 3 \) and find \( x^6 + \frac{1}{x^6} \), we can follow these steps:

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

We start with the equation:
\[ x + \frac{1}{x} = 3 \]

We can square both sides to find \( x^2 + \frac{1}{x^2} \):
\[ \left( x + \frac{1}{x} \right)^2 = 3^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 9 \]
Subtracting 2 from both sides gives:
\[ x^2 + \frac{1}{x^2} = 9 - 2 = 7 \]

### Step 2: Find \( x^4 + \frac{1}{x^4} \)

Next, we square \( x^2 + \frac{1}{x^2} \) to find \( x^4 + \frac{1}{x^4} \):
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 7^2 \]
This expands to:
\[ x^4 + 2 + \frac{1}{x^4} = 49 \]
Subtracting 2 from both sides gives:
\[ x^4 + \frac{1}{x^4} = 49 - 2 = 47 \]

### Step 3: Find \( x^6 + \frac{1}{x^6} \)

Now, we can find \( x^6 + \frac{1}{x^6} \) using the relation:
\[ x^6 + \frac{1}{x^6} = \left( x^4 + \frac{1}{x^4} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x^2 + \frac{1}{x^2} \right) \]
Substituting the values we found:
\[ x^6 + \frac{1}{x^6} = (47)(7) - 7 \]
Calculating \( 47 \times 7 \):
\[ 47 \times 7 = 329 \]
Thus:
\[ x^6 + \frac{1}{x^6} = 329 - 7 = 322 \]

### Final Answer
\[ x^6 + \frac{1}{x^6} = 322 \]

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