`(x−y)(x+y)(x^2+y^2)(x^4+y^4)` is equal to=    

Options

A. `x^16 − y^16`
B. `x^8 − y^8`
C. `x^8 + y^8`
D. `x^16 + y^16`

Answer

To solve the expression \((x−y)(x+y)(x^2+y^2)(x^4+y^4)\), we can apply algebraic identities step by step.

### Step 1: Simplify \((x - y)(x + y)\)
Using the identity \(a - b)(a + b = a^2 - b^2\), we can simplify the first part:
\[ (x - y)(x + y) = x^2 - y^2 \]

### Step 2: Rewrite the expression
Now, we can rewrite the original expression using the result from Step 1:
\[ (x^2 - y^2)(x^2 + y^2)(x^4 + y^4) \]

### Step 3: Simplify \((x^2 - y^2)(x^2 + y^2)\)
Again, we can use the same identity \(a - b)(a + b = a^2 - b^2\) on the new expression:
\[ (x^2 - y^2)(x^2 + y^2) = x^4 - y^4 \]

### Step 4: Rewrite the expression again
Now, we can rewrite the expression:
\[ (x^4 - y^4)(x^4 + y^4) \]

### Step 5: Simplify \((x^4 - y^4)(x^4 + y^4)\)
Once more, we apply the identity:
\[ (x^4 - y^4)(x^4 + y^4) = x^8 - y^8 \]

### Final Result
Thus, the final result of the expression \((x−y)(x+y)(x^2+y^2)(x^4+y^4)\) is:
\[ \boxed{x^8 - y^8} \]