If `(x + 1/x) = sqrt5`, then find the value of `x^4 + 1/x^4`

Options

A. 8
B. 3
C. 5
D. 7

Answer

To solve the problem, we need to find the value of \( x^4 + \frac{1}{x^4} \) given that \( x + \frac{1}{x} = \sqrt{5} \).

### Step-by-Step Solution:

1. **Given Equation**:
\[ x + \frac{1}{x} = \sqrt{5} \]

2. **Square Both Sides**:
To eliminate the fraction, we square both sides:
\[ \left( x + \frac{1}{x} \right)^2 = (\sqrt{5})^2 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = 5 \]

3. **Rearranging the Equation**:
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 5 - 2 \]
Thus,
\[ x^2 + \frac{1}{x^2} = 3 \]

4. **Square Again**:
Next, we need to find \( x^4 + \frac{1}{x^4} \). We can do this by squaring \( x^2 + \frac{1}{x^2} \):
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 3^2 \]
This expands to:
\[ x^4 + 2 + \frac{1}{x^4} = 9 \]

5. **Rearranging Again**:
Now, we can isolate \( x^4 + \frac{1}{x^4} \):
\[ x^4 + \frac{1}{x^4} = 9 - 2 \]
Thus,
\[ x^4 + \frac{1}{x^4} = 7 \]

### Final Answer:
The value of \( x^4 + \frac{1}{x^4} \) is \( 7 \).

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