If `x+ (1)/(x)=3`, then
`x^(4) + (1)/(x^(4))`= ?

Answer

To solve the equation \( x + \frac{1}{x} = 3 \) and find \( x^4 + \frac{1}{x^4} \), we can follow these steps:

### Step 1: Square both sides of the equation
Starting with the given equation:
\[ x + \frac{1}{x} = 3 \]
We square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = 3^2 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = 9 \]

### Step 2: Rearrange to find \( x^2 + \frac{1}{x^2} \)
From the equation obtained in Step 1:
\[ x^2 + \frac{1}{x^2} + 2 = 9 \]
We can isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 9 - 2 = 7 \]

### Step 3: Square \( x^2 + \frac{1}{x^2} \)
Now we need to find \( x^4 + \frac{1}{x^4} \). We can use the identity:
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + 2 + \frac{1}{x^4} \]
Substituting the value we found:
\[ 7^2 = x^4 + 2 + \frac{1}{x^4} \]
This simplifies to:
\[ 49 = x^4 + 2 + \frac{1}{x^4} \]

### Step 4: Rearrange to find \( x^4 + \frac{1}{x^4} \)
Now, we can isolate \( x^4 + \frac{1}{x^4} \):
\[ x^4 + \frac{1}{x^4} = 49 - 2 = 47 \]

### Final Answer
Thus, the value of \( x^4 + \frac{1}{x^4} \) is:
\[ \boxed{47} \]

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