Question

\frac{ x^{2}y^{2} }{ x^{2} - y^{2} } при х= \frac{2ab}{ a^{2}- b^{2} } при у= \frac{2ab}{ a^{2}+ b^{2} }

Answer 1

$\frac{ x^{2}y^{2} }{ x^{2} - y^{2} }=\frac{ x^{2}y^{2} }{ (x - y)(x + y) }=\frac{ (\frac{2ab}{ a^{2}- b^{2} })^{2}(\frac{2ab}{ a^{2}+ b^{2} })^{2} }{ (\frac{2ab}{ a^{2}- b^{2} } - \frac{2ab}{ a^{2}+ b^{2} })(\frac{2ab}{ a^{2}- b^{2} } + \frac{2ab}{ a^{2}+ b^{2} }) }= \\ =\frac{ \frac{16a^4b^4}{ a^4- b^4 } }{ 4a^2b^2(\frac{a^{2}+ b^{2}-(a^{2}- b^{2})}{ (a^{2}- b^{2})(a^{2}+ b^{2}) } )(\frac{a^{2}+ b^{2}+a^{2}- b^{2}}{ (a^{2}- b^{2})(a^{2}+ b^{2}) }) }=$

$=\frac{ 4a^2b^2 }{ 2b^{2} (\frac{2a^{2}}{ (a^{2}- b^{2})(a^{2}+ b^{2}) }) }=(a^{2}- b^{2})(a^{2}+ b^{2})=a^4- b^4$

Answer 2

It looks like you've provided an expression involving variables x and y, and specific values for x and y. Let's break down the expression and evaluate it step by step.

Given expression: \frac{x^{2}y^{2}}{x^{2} - y^{2}}

Values: x = \frac{2ab}{a^{2} - b^{2}} y = \frac{2ab}{a^{2} + b^{2}}

First, let's substitute the given values of x and y into the expression: \frac{\left(\frac{2ab}{a^{2} - b^{2}}\right)^{2} \left(\frac{2ab}{a^{2} + b^{2}}\right)^{2}}{\left(\frac{2ab}{a^{2} - b^{2}}\right)^{2} - \left(\frac{2ab}{a^{2} + b^{2}}\right)^{2}}

Now, let's simplify the expression step by step.

Substitute values of x and y

\frac{\left(\frac{2ab}{a^{2} - b^{2}}\right)^{2} \left(\frac{2ab}{a^{2} + b^{2}}\right)^{2}}{\left(\frac{2ab}{a^{2} - b^{2}}\right)^{2} - \left(\frac{2ab}{a^{2} + b^{2}}\right)^{2}}

Simplify the numerator

\frac{\left(\frac{4a^{2}b^{2}}{(a^{2} - b^{2})^{2}}\right) \left(\frac{4a^{2}b^{2}}{(a^{2} + b^{2})^{2}}\right)}{\left(\frac{4a^{2}b^{2}}{(a^{2} - b^{2})^{2}}\right) - \left(\frac{4a^{2}b^{2}}{(a^{2} + b^{2})^{2}}\right)}

Multiply the terms in the numerator

\frac{\frac{16a^{4}b^{4}}{(a^{2} - b^{2})^{2}(a^{2} + b^{2})^{2}}}{\frac{4a^{2}b^{2}(a^{2} + b^{2})^{2} - 4a^{2}b^{2}(a^{2} - b^{2})^{2}}{(a^{2} - b^{2})^{2}(a^{2} + b^{2})^{2}}}

Simplify the denominator

\frac{\frac{16a^{4}b^{4}}{(a^{2} - b^{2})^{2}(a^{2} + b^{2})^{2}}}{\frac{4a^{2}b^{2}(a^{2} + b^{2})^{2} - 4a^{2}b^{2}(a^{2} - b^{2})^{2}}{(a^{2} - b^{2})^{2}(a^{2} + b^{2})^{2}}} \frac{\frac{16a^{4}b^{4}}{(a^{2} - b^{2})^{2}(a^{2} + b^{2})^{2}}}{4a^{2}b^{2(a^{2} + b^{2})^{2} - (a^{2} - b^{2})^{2}}}

At this point, the expression has been simplified as much as possible without specific values for a and b. If you have specific values for a and b, you can substitute them in to further simplify the expression. Let me know if you need further assistance with this!