Find the square root of
`1+a^(2)+sqrt(1+a^(2)+a^(4))`

Answer

To find the square root of the expression \( 1 + a^2 + \sqrt{1 + a^2 + a^4} \), we can follow these steps:

### Step 1: Simplify the inner square root
We start with the expression:
\[ \sqrt{1 + a^2 + a^4} \]
We can rewrite \( a^4 \) as \( (a^2)^2 \), so we have:
\[ \sqrt{1 + a^2 + (a^2)^2} = \sqrt{1 + a^2 + a^4} \]

### Step 2: Recognize the quadratic form
The expression \( 1 + a^2 + a^4 \) can be seen as a quadratic in terms of \( a^2 \):
\[ a^4 + a^2 + 1 \]
This can be factored or simplified further, but we will compute directly.

### Step 3: Substitute back into the original expression
Now, we substitute this back into the original expression:
\[ 1 + a^2 + \sqrt{1 + a^2 + a^4} \]

### Step 4: Combine terms
We can combine the terms:
Let \( x = \sqrt{1 + a^2 + a^4} \), then we have:
\[ 1 + a^2 + x \]

### Step 5: Find the square root of the entire expression
Now we need to find:
\[ \sqrt{1 + a^2 + x} \]

### Step 6: Use the identity for simplification
Using the identity \( (A + B)^2 = A^2 + 2AB + B^2 \), we can express \( 1 + a^2 + x \) in a form that can be squared:
Let \( A = \sqrt{1 + a^2} \) and \( B = \sqrt{x} \), then:
\[ 1 + a^2 + x = A^2 + B^2 + 2AB \]

### Step 7: Final computation
Thus, we can express the square root as:
\[ \sqrt{(A + B)^2} = A + B \]
This gives us:
\[ \sqrt{1 + a^2 + \sqrt{1 + a^2 + a^4}} = \sqrt{1 + a^2} + \sqrt{\sqrt{1 + a^2 + a^4}} \]

### Conclusion
The final result can be simplified to:
\[ \sqrt{1 + a^2} + \sqrt{1 + a^2 + a^4} \]