What is the simplified value of
` ( x + (1)/( x)) ( x^(2) + (1)/( x^(2))) ( x^(4) + (1) /( x^(4))) ( x^(8) + (1)/( x^(6))) ( x^(16) + (1)/( x^(16)))`

Options

A. `(( x ^(654) - (1)/( x^(64))))/(( x - (1)/( x)))`
B. `(( x ^(8) - (1)/( x^(8))))/(( x - (1)/( x)))`
C. `(( x ^(16) - (1)/( x^(16))))/(( x - (1)/( x)))`
D. `(( x ^(32) - (1)/( x^(32))))/(( x - (1)/( x)))`

Answer

To simplify the expression

\[ (x + \frac{1}{x})(x^2 + \frac{1}{x^2})(x^4 + \frac{1}{x^4})(x^8 + \frac{1}{x^8})(x^{16} + \frac{1}{x^{16}}) \]

we can follow these steps:

### Step 1: Rewrite Each Term
We can rewrite each term in the product using the identity \( a + \frac{1}{a} = \frac{a^2 + 1}{a} \).

1. \( x + \frac{1}{x} = \frac{x^2 + 1}{x} \)
2. \( x^2 + \frac{1}{x^2} = \frac{x^4 + 1}{x^2} \)
3. \( x^4 + \frac{1}{x^4} = \frac{x^8 + 1}{x^4} \)
4. \( x^8 + \frac{1}{x^8} = \frac{x^{16} + 1}{x^8} \)
5. \( x^{16} + \frac{1}{x^{16}} = \frac{x^{32} + 1}{x^{16}} \)

### Step 2: Substitute Back into the Expression
Substituting these back into the original expression gives:

\[ \left(\frac{x^2 + 1}{x}\right) \left(\frac{x^4 + 1}{x^2}\right) \left(\frac{x^8 + 1}{x^4}\right) \left(\frac{x^{16} + 1}{x^8}\right) \left(\frac{x^{32} + 1}{x^{16}}\right) \]

### Step 3: Simplify the Expression
Now we can multiply these fractions together. The numerator will be:

\[ (x^2 + 1)(x^4 + 1)(x^8 + 1)(x^{16} + 1)(x^{32} + 1) \]

And the denominator will be:

\[ x \cdot x^2 \cdot x^4 \cdot x^8 \cdot x^{16} = x^{1 + 2 + 4 + 8 + 16} = x^{31} \]

Thus, the expression simplifies to:

\[ \frac{(x^2 + 1)(x^4 + 1)(x^8 + 1)(x^{16} + 1)(x^{32} + 1)}{x^{31}} \]

### Step 4: Final Simplification
The final simplified expression is:

\[ \frac{(x^2 + 1)(x^4 + 1)(x^8 + 1)(x^{16} + 1)(x^{32} + 1)}{x^{31}} \]