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(2^x+2^(-x))^2(2x-2^(-x))^2 1(3^x+3^(-x))^2(3^x-3^(-x))^2 1(4^x+4^(-x))^2(4^x-4^(-x))1

3 min read Jun 16, 2024
(2^x+2^(-x))^2(2x-2^(-x))^2 1(3^x+3^(-x))^2(3^x-3^(-x))^2 1(4^x+4^(-x))^2(4^x-4^(-x))1

Simplifying Complex Exponential Expressions

This article will explore the simplification of the following complex exponential expression:

(2^x + 2^(-x))^2 (2^x - 2^(-x))^2 (3^x + 3^(-x))^2 (3^x - 3^(-x))^2 (4^x + 4^(-x))^2 (4^x - 4^(-x))^2

We can effectively simplify this expression by utilizing the following key algebraic identities:

1. Difference of Squares: a² - b² = (a + b)(a - b) 2. Perfect Square Trinomial: (a + b)² = a² + 2ab + b²

Let's break down the simplification step-by-step:

Step 1: Applying the Difference of Squares Identity

  • Notice that each pair of consecutive terms in the expression has the form (a + b)(a - b). Applying the difference of squares identity, we can simplify each pair:

    • (2^x + 2^(-x))(2^x - 2^(-x)) = (2^x)² - (2^(-x))² = 2^(2x) - 2^(-2x)
    • (3^x + 3^(-x))(3^x - 3^(-x)) = (3^x)² - (3^(-x))² = 3^(2x) - 3^(-2x)
    • (4^x + 4^(-x))(4^x - 4^(-x)) = (4^x)² - (4^(-x))² = 4^(2x) - 4^(-2x)

Step 2: Combining the Simplified Terms

  • Our expression now becomes:

(2^(2x) - 2^(-2x)) (3^(2x) - 3^(-2x)) (4^(2x) - 4^(-2x))

Step 3: Factoring by Grouping (Optional)

  • This step is not strictly necessary for simplification, but it can provide a more compact form. We can factor by grouping:

    • Group 1: (2^(2x) - 2^(-2x)) (3^(2x) - 3^(-2x))
    • Group 2: (4^(2x) - 4^(-2x))

  • Apply the difference of squares identity again within each group. However, we will omit this step for brevity.

Step 4: Final Simplified Form

  • The final simplified form of the expression is:

[(2^(2x) - 2^(-2x)) (3^(2x) - 3^(-2x))] [(4^(2x) - 4^(-2x))]

  • This expression is significantly simpler than the original and is easier to manipulate for further calculations or analysis.

Conclusion

By effectively applying algebraic identities, we have successfully simplified a complex exponential expression. This process highlights the power of algebraic manipulation in simplifying complex expressions and making them more manageable for further operations.

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