What is the simplified value of `(x^(128)+1)(x^(32)+1)(x^(64)+1)(x^(16)+1)(x^(8)+1)(x^(4)+1)(x^(2)+1)(x+1)` ?

Options

A. `x^(256)-1`
B. `(x^(128)-1)/(x-1)`
C. `(x^(64)-1)/(x-1)`
D. `(x^(256)-1)/(x-1)`

Answer

To simplify the expression \((x^{128}+1)(x^{64}+1)(x^{32}+1)(x^{16}+1)(x^{8}+1)(x^{4}+1)(x^{2}+1)(x+1)\), we can follow these steps:

### Step 1: Recognize the Pattern
Notice that the expression consists of terms of the form \(x^{2^n} + 1\) for \(n = 0, 1, 2, 3, 4, 5, 6, 7\).

### Step 2: Use the Identity
We can use the identity:
\[ x^{2^n} + 1 = \frac{x^{2^{n+1}} - 1}{x^{2^n} - 1} \]
This means that we can express \(x^{2^n} + 1\) in terms of \(x^{2^{n+1}} - 1\).

### Step 3: Multiply the Terms
We can multiply all the terms together:
\[ (x^{128}+1)(x^{64}+1)(x^{32}+1)(x^{16}+1)(x^{8}+1)(x^{4}+1)(x^{2}+1)(x+1) \]
Using the identity, we can rewrite this as:
\[ \frac{x^{256}-1}{x^{128}-1} \cdot \frac{x^{128}-1}{x^{64}-1} \cdot \frac{x^{64}-1}{x^{32}-1} \cdots \frac{x^{4}-1}{x^{2}-1} \cdot (x+1) \]

### Step 4: Cancel Terms
Notice that all intermediate terms will cancel out:
\[ = \frac{x^{256}-1}{x-1} \]

### Step 5: Final Expression
Thus, the simplified value of the original expression is:
\[ x^{256}-1 \]

### Final Answer
The simplified value is \(x^{256}-1\).

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