Value of `sqrt(1 + x^2 + sqrt(1 + x^2 + x^4))` is

Options

A. `(1)/(sqrt(2)) (sqrt(1 + x + x^2) + sqrt(1 - x + x^2))`
B. `1/(sqrt(2)) (sqrt(1 + x + x^2) - sqrt(1 - x + x^2))`
C. `1/(sqrt2)(sqrt(1 + x + x^2) - sqrt(1 + x - x^2))`
D. None of these

Answer

To solve the expression \( \sqrt{1 + x^2 + \sqrt{1 + x^2 + x^4}} \), we will follow these steps:

### Step 1: Simplify the inner square root
We start with the expression:
\[ \sqrt{1 + x^2 + \sqrt{1 + x^2 + x^4}} \]
First, we simplify the inner square root \( \sqrt{1 + x^2 + x^4} \).

Notice that:
\[ 1 + x^2 + x^4 = (x^2 + 1)^2 \]
Thus, we can rewrite the inner square root:
\[ \sqrt{1 + x^2 + x^4} = \sqrt{(x^2 + 1)^2} = x^2 + 1 \]

### Step 2: Substitute back into the expression
Now we substitute this back into the original expression:
\[ \sqrt{1 + x^2 + (x^2 + 1)} = \sqrt{1 + x^2 + x^2 + 1} = \sqrt{2 + 2x^2} \]

### Step 3: Factor out the common term
We can factor out the 2 from the expression under the square root:
\[ \sqrt{2(1 + x^2)} \]

### Step 4: Simplify the square root
Using the property of square roots, we can simplify this further:
\[ \sqrt{2(1 + x^2)} = \sqrt{2} \cdot \sqrt{1 + x^2} \]

### Final Result
Thus, the value of \( \sqrt{1 + x^2 + \sqrt{1 + x^2 + x^4}} \) is:
\[ \sqrt{2} \cdot \sqrt{1 + x^2} \]