If ` ( x+ (1)/( x)) = 3 then ( x^(8) + (1)/( x^(8)))` is equal to

Options

A. 2201
B. 2203
C. 2207
D. 2213

Answer

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

\[ x + \frac{1}{x} = 3 \]

We need to find \(x^8 + \frac{1}{x^8}\). To do this, we will use a systematic approach by finding \(x^2 + \frac{1}{x^2}\), \(x^4 + \frac{1}{x^4}\), and finally \(x^8 + \frac{1}{x^8}\).

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

We can square both sides of the equation \(x + \frac{1}{x} = 3\):

\[ \left(x + \frac{1}{x}\right)^2 = 3^2 \]

This expands to:

\[ x^2 + 2 + \frac{1}{x^2} = 9 \]

Now, we can isolate \(x^2 + \frac{1}{x^2}\):

\[ x^2 + \frac{1}{x^2} = 9 - 2 = 7 \]

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

Next, we square \(x^2 + \frac{1}{x^2}\):

\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 7^2 \]

This expands to:

\[ x^4 + 2 + \frac{1}{x^4} = 49 \]

Isolating \(x^4 + \frac{1}{x^4}\):

\[ x^4 + \frac{1}{x^4} = 49 - 2 = 47 \]

### Step 3: Find \(x^8 + \frac{1}{x^8}\)

Now, we square \(x^4 + \frac{1}{x^4}\):

\[ \left(x^4 + \frac{1}{x^4}\right)^2 = 47^2 \]

This expands to:

\[ x^8 + 2 + \frac{1}{x^8} = 2209 \]

Isolating \(x^8 + \frac{1}{x^8}\):

\[ x^8 + \frac{1}{x^8} = 2209 - 2 = 2207 \]

### Final Answer

Thus, the value of \(x^8 + \frac{1}{x^8}\) is:

\[ \boxed{2207} \]