(x%2By)(x%5E2%2By%5E2)(x%5E4%2By%5E4)(x%5E8%2By%5E8).jpeg "(x-y)(x+y)(x^2+y^2)(x^4+y^4)(x^8+y^8)")
Unraveling the Mystery: (x-y)(x+y)(x^2+y^2)(x^4+y^4)(x^8+y^8)
At first glance, the expression (x-y)(x+y)(x^2+y^2)(x^4+y^4)(x^8+y^8) might seem daunting. However, with a little bit of algebraic manipulation, it simplifies beautifully. Let's break it down step by step.
The Power of Difference of Squares
The key to simplifying this expression lies in recognizing the difference of squares pattern: (a - b)(a + b) = a² - b².
- Notice that the first two terms, (x-y)(x+y), directly fit this pattern. Expanding them, we get x² - y².
Expanding the Expression
Now our expression becomes: (x² - y²)(x² + y²)(x⁴ + y⁴)(x⁸ + y⁸)
- Again, the first two terms, (x² - y²)(x² + y²), follow the difference of squares pattern. Expanding them, we obtain x⁴ - y⁴.
Our expression now reads: (x⁴ - y⁴)(x⁴ + y⁴)(x⁸ + y⁸)
- Once again, we can apply the difference of squares pattern to the first two terms. This leads to x⁸ - y⁸.
The final step: (x⁸ - y⁸)(x⁸ + y⁸)
- Applying the difference of squares pattern one last time, we arrive at our simplified answer: x¹⁶ - y¹⁶
Conclusion
The seemingly complex expression (x-y)(x+y)(x^2+y^2)(x^4+y^4)(x^8+y^8) simplifies elegantly to x¹⁶ - y¹⁶ through repeated application of the difference of squares pattern. This demonstrates how recognizing and utilizing algebraic patterns can significantly simplify complex mathematical expressions.