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Solving the Value of a Complex Expression: 2^{(x+y)^{2}}/2^{(x-y)^{2}} When xy = 1

Mathematics can be both fascinating and frustrating, especially when dealing with complex expressions that seem to defy simplicity. However, there are often hidden patterns and tricks that can simplify seemingly complicated problems, and it’s often these moments of enlightenment that make math worth the effort. In this post, we will explore how to solve the value of a complex expression involving exponents and variables, under a specific condition that relates these variables in a non-obvious way. Specifically, we will consider the expression

E = 2^{(x+y)^{2}}/2^{(x-y)^{2}}

when xy = 1. This expression may look intimidating at first, but we can make some observations and substitutions that will lead us to an elegant solution.

First, let’s try to expand and simplify the numerator and denominator of the fraction separately. We use the property that (a+b)^2 = a^2 + 2ab + b^2 and (a-b)^2 = a^2 – 2ab + b^2, and the fact that 1/2 = 2^{-1}, to obtain

E = 2^{(x+y)^{2}}/2^{(x-y)^{2}} = 2^{(x^2+2xy+y^2)} * 2^{-(x^2-2xy+y^2)} = 2^{4xy} * 2^{4y^2} / 2^{4x^2} = 16y^{2}/16x^{2}

where we used the fact that xy = 1 implies x^2y^2 = 1 and thus 4xy = 4/x^2 = 4y^2. Now we can simplify this expression further by canceling out the common factor of 16 and dividing y^2 by x^2:

E = (y/x)^{2} = x^{2}/y^{2}

This means that the value of the expression E depends solely on the values of x and y, not on any other variables or parameters. In particular, if we know that xy = 1, we can substitute y = 1/x into the expression E and obtain

E = x^2/(1/x)^2 = x^4

This result is both simple and surprising. It tells us that the original complex expression E reduces to the fourth power of x when xy = 1, regardless of the value of y. It also shows us that the exponent laws are powerful tools that enable us to manipulate exponents in various ways, and that the connections between seemingly unrelated variables can be used to simplify complex expressions.

To summarize, we have solved the value of the complex expression 2^{(x+y)^{2}}/2^{(x-y)^{2}} when xy = 1 by expanding and simplifying the numerator and denominator separately, canceling out common factors, and substituting y = 1/x. We found that the expression simplifies to x^4, which depends only on the value of x and is independent of y. We also learned some general techniques for handling exponents and using algebraic relations between variables. Hopefully, this example illustrates the beauty and utility of mathematics, and encourages you to explore more of its mysteries and wonders.