Simplify `:sqrt(1+ ((x ^(4) -1 )/( -2 x ^(2)) ) ^(2))`

Options

A. `(sqrt (x ^(2) + 1 ))/( 2 x ^(2))`
B. `(x ^(4) + 2x ^(2) +1 )/(2x ^(2))`
C. `(x ^(4) -1)/(2x ^(2))`
D. `(x ^(4) + 1)/( 2x ^(2))`

Answer

To simplify the expression \( \sqrt{1 + \left( \frac{x^4 - 1}{-2x^2} \right)^2} \), we will follow these steps:

### Step 1: Rewrite the expression inside the square root
We start with the expression:
\[ \sqrt{1 + \left( \frac{x^4 - 1}{-2x^2} \right)^2} \]
Calculating the square of the fraction:
\[ \left( \frac{x^4 - 1}{-2x^2} \right)^2 = \frac{(x^4 - 1)^2}{(2x^2)^2} = \frac{(x^4 - 1)^2}{4x^4} \]
Thus, we can rewrite the expression as:
\[ \sqrt{1 + \frac{(x^4 - 1)^2}{4x^4}} \]

### Step 2: Combine the terms under a common denominator
To combine the terms under the square root, we need a common denominator:
\[ 1 = \frac{4x^4}{4x^4} \]
So we have:
\[ \sqrt{\frac{4x^4 + (x^4 - 1)^2}{4x^4}} \]

### Step 3: Expand \((x^4 - 1)^2\)
Now, we expand \((x^4 - 1)^2\):
\[ (x^4 - 1)^2 = x^8 - 2x^4 + 1 \]
Substituting this back into our expression gives:
\[ \sqrt{\frac{4x^4 + x^8 - 2x^4 + 1}{4x^4}} = \sqrt{\frac{x^8 + 2x^4 + 1}{4x^4}} \]

### Step 4: Factor the numerator
Notice that the numerator can be factored:
\[ x^8 + 2x^4 + 1 = (x^4 + 1)^2 \]
Thus, we can rewrite the expression as:
\[ \sqrt{\frac{(x^4 + 1)^2}{4x^4}} \]

### Step 5: Simplify the square root
Now we can simplify the square root:
\[ \sqrt{\frac{(x^4 + 1)^2}{4x^4}} = \frac{x^4 + 1}{2x^2} \]

### Final Result
The simplified expression is:
\[ \frac{x^4 + 1}{2x^2} \]