)%5E2(2x-2%5E(-x))%5E2-1(3%5Ex%2B3%5E(-x))%5E2(3%5Ex-3%5E(-x))%5E2-1(4%5Ex%2B4%5E(-x))%5E2(4%5Ex-4%5E(-x))1.jpeg "(2x+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 a complex expression involving exponential terms. The expression in question is:
(2x + 2^(-x))² (2x - 2^(-x))² (3^x + 3^(-x))² (3^x - 3^(-x))² (4^x + 4^(-x))² (4^x - 4^(-x))
Let's break down the simplification process step-by-step.
Utilizing Algebraic Identities
The key to simplifying this expression lies in recognizing and applying algebraic identities. Here are the key identities we will utilize:
- Difference of Squares: (a + b)(a - b) = a² - b²
- Square of a Sum: (a + b)² = a² + 2ab + b²
- Square of a Difference: (a - b)² = a² - 2ab + b²
Simplifying the Expression
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Focus on the first two terms: (2x + 2^(-x))² (2x - 2^(-x))² can be simplified using the difference of squares identity: (2x + 2^(-x))² (2x - 2^(-x))² = [(2x)² - (2^(-x))²]² = (4x² - 2^(-2x))²
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Simplifying the next two terms: (3^x + 3^(-x))² (3^x - 3^(-x))² can also be simplified using the difference of squares identity: (3^x + 3^(-x))² (3^x - 3^(-x))² = [(3^x)² - (3^(-x))²]² = (9^x - 3^(-2x))²
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Simplifying the last two terms: (4^x + 4^(-x))² (4^x - 4^(-x)) can be simplified using the difference of squares identity: (4^x + 4^(-x))² (4^x - 4^(-x)) = [(4^x)² - (4^(-x))²] = (16^x - 4^(-2x))
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Combining the results: The simplified expression becomes: (4x² - 2^(-2x))² (9^x - 3^(-2x))² (16^x - 4^(-2x))
Further Simplification (Optional)
While the expression is now simplified, it can be further manipulated if desired. For example, you could:
- Expand the squares: Using the square of a binomial identity, you could expand each of the squared terms.
- Factor out common terms: If there are common factors within the expression, you could factor them out for a more concise representation.
Conclusion
By strategically applying algebraic identities, we successfully simplified a complex exponential expression. The resulting simplified expression offers a clearer representation of the original complex formula, making it easier to analyze or use in further calculations. Remember to utilize the appropriate identities and always be mindful of the potential for further simplification.