Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
(y/(x-y)+x/(x+y))/(1/x^2+1/y^2)-y^4/(x^2-y^2)
How do you (y/(x-y)+x/(x+y))/(1/x^2+1/y^2)-y^4/(x^2-y^2) in partial fractions?
The solution
You have entered
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y x
----- + ----- 4
x - y x + y y
------------- - -------
1 1 2 2
-- + -- x - y
2 2
x y
$$- \frac{y^{4}}{x^{2} - y^{2}} + \frac{\frac{x}{x + y} + \frac{y}{x - y}}{\frac{1}{y^{2}} + \frac{1}{x^{2}}}$$
(y/(x - y) + x/(x + y))/(1/(x^2) + 1/(y^2)) - y^4/(x^2 - y^2)
Rational denominator
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2 2 / 2 2\ 4 / 2 2\
x *y *\x - y /*(x*(x - y) + y*(x + y)) - y *(x + y)*(x - y)*\x + y /
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/ 2 2\ / 2 2\
(x + y)*(x - y)*\x + y /*\x - y /
$$\frac{x^{2} y^{2} \left(x^{2} - y^{2}\right) \left(x \left(x - y\right) + y \left(x + y\right)\right) - y^{4} \left(x - y\right) \left(x + y\right) \left(x^{2} + y^{2}\right)}{\left(x - y\right) \left(x + y\right) \left(x^{2} - y^{2}\right) \left(x^{2} + y^{2}\right)}$$
(x^2*y^2*(x^2 - y^2)*(x*(x - y) + y*(x + y)) - y^4*(x + y)*(x - y)*(x^2 + y^2))/((x + y)*(x - y)*(x^2 + y^2)*(x^2 - y^2))
Numerical answer
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(x/(x + y) + y/(x - y))/(x^(-2) + y^(-2)) - y^4/(x^2 - y^2)
(x/(x + y) + y/(x - y))/(x^(-2) + y^(-2)) - y^4/(x^2 - y^2)
Combining rational expressions
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2 / 2 / 2 2\ 2 / 2 2\\
y *\x *\x - y /*(x*(x - y) + y*(x + y)) - y *(x + y)*(x - y)*\x + y //
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/ 2 2\ / 2 2\
(x + y)*(x - y)*\x + y /*\x - y /
$$\frac{y^{2} \left(x^{2} \left(x^{2} - y^{2}\right) \left(x \left(x - y\right) + y \left(x + y\right)\right) - y^{2} \left(x - y\right) \left(x + y\right) \left(x^{2} + y^{2}\right)\right)}{\left(x - y\right) \left(x + y\right) \left(x^{2} - y^{2}\right) \left(x^{2} + y^{2}\right)}$$
y^2*(x^2*(x^2 - y^2)*(x*(x - y) + y*(x + y)) - y^2*(x + y)*(x - y)*(x^2 + y^2))/((x + y)*(x - y)*(x^2 + y^2)*(x^2 - y^2))