What is the simplified value of `( x^(32) + (1)/(x^(32)))( x^(8) + (1)/( z^(8))) (x - (1)/( x)) ( x^(16) + (1)/( x^(16))) ( x + (1)/(x)) ( x^(4) + (1)/( x^(4)))`

Options

A. `(x 64 + (1)/( x^(64)))`
B. `((x^(64)- (1)/( x^(64))))/((x^(2) + (1)/( x^(2))))`
C. `(( x^(64) -(1)/( x^(64))))/(( x + (1)/( x)))`
D. `(( x^(32) - (1)/( x^(64))))/( ( x + (1)/( x)))`

Answer

To simplify the expression

\[ (x^{32} + \frac{1}{x^{32}})(x^{8} + \frac{1}{x^{8}})(x - \frac{1}{x})(x^{16} + \frac{1}{x^{16}})(x + \frac{1}{x})(x^{4} + \frac{1}{x^{4}}), \]

we can follow these steps:

### Step 1: Simplifying the expression \( (x - \frac{1}{x}) \)
We recognize that \( (x - \frac{1}{x}) \) can be rewritten in terms of \( (x + \frac{1}{x}) \):
\[ (x - \frac{1}{x})(x + \frac{1}{x}) = x^2 - \frac{1}{x^2}. \]

### Step 2: Combine \( (x + \frac{1}{x}) \) and \( (x - \frac{1}{x}) \)
Using the result from Step 1, we can rewrite:
\[ (x - \frac{1}{x})(x + \frac{1}{x}) = x^2 - \frac{1}{x^2}. \]

### Step 3: Simplifying \( (x^4 + \frac{1}{x^4}) \)
Next, we can use the identity:
\[ x^4 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2. \]

### Step 4: Simplifying \( (x^8 + \frac{1}{x^8}) \)
Similarly, we can use:
\[ x^8 + \frac{1}{x^8} = (x^4 + \frac{1}{x^4})^2 - 2. \]

### Step 5: Simplifying \( (x^{16} + \frac{1}{x^{16}}) \)
Continuing with the same pattern:
\[ x^{16} + \frac{1}{x^{16}} = (x^8 + \frac{1}{x^8})^2 - 2. \]

### Step 6: Simplifying \( (x^{32} + \frac{1}{x^{32}}) \)
Finally, we apply the same identity:
\[ x^{32} + \frac{1}{x^{32}} = (x^{16} + \frac{1}{x^{16}})^2 - 2. \]

### Step 7: Putting it all together
Now we can combine all these results:
\[ (x^{32} + \frac{1}{x^{32}})(x^{8} + \frac{1}{x^{8}})(x^{16} + \frac{1}{x^{16}})(x^{4} + \frac{1}{x^{4}}) = \text{(combined results)}. \]

### Final Result
After all the simplifications, we can express the entire product in terms of \( x^{64} - \frac{1}{x^{64}} \) divided by \( x^2 + \frac{1}{x^2} \):
\[ \frac{x^{64} - \frac{1}{x^{64}}}{x^2 + \frac{1}{x^2}}. \]