Mister Exam
Least common denominator ((x/(y-x))^-2)*((x^2+x*y)/(((x+y)^2)+4*x*y))
The solution
You have entered
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/ 2 \
| x + x*y |
|----------------|
| 2 |
\(x + y) + 4*x*y/
------------------
2
/ x \
|-----|
\y - x/
$$\frac{\left(x^{2} + x y\right) \frac{1}{4 x y + \left(x + y\right)^{2}}}{x^{2} \frac{1}{\left(- x + y\right)^{2}}}$$
((x^2 + x*y)/((x + y)^2 + (4*x)*y))/(x/(y - x))^2
General simplification
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2 (x - y) *(x + y) -------------------- / 2 \ x*\(x + y) + 4*x*y/
$$\frac{\left(x - y\right)^{2} \left(x + y\right)}{x \left(4 x y + \left(x + y\right)^{2}\right)}$$
(x - y)^2*(x + y)/(x*((x + y)^2 + 4*x*y))
Assemble expression
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2 / 2 \ (y - x) *\x + x*y/ --------------------- 2 / 2 \ x *\(x + y) + 4*x*y/
$$\frac{\left(- x + y\right)^{2} \left(x^{2} + x y\right)}{x^{2} \left(4 x y + \left(x + y\right)^{2}\right)}$$
(y - x)^2*(x^2 + x*y)/(x^2*((x + y)^2 + 4*x*y))
Numerical answer
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(y - x)^2*(x^2 + x*y)/(x^2*((x + y)^2 + 4.0*x*y))
(y - x)^2*(x^2 + x*y)/(x^2*((x + y)^2 + 4.0*x*y))
Trigonometric part
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2 / 2 \ (y - x) *\x + x*y/ --------------------- 2 / 2 \ x *\(x + y) + 4*x*y/
$$\frac{\left(- x + y\right)^{2} \left(x^{2} + x y\right)}{x^{2} \left(4 x y + \left(x + y\right)^{2}\right)}$$
(y - x)^2*(x^2 + x*y)/(x^2*((x + y)^2 + 4*x*y))
Rational denominator
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2 2 2
x *(x - y) + x*y*(x - y)
--------------------------
2 / 2 2 \
x *\x + y + 6*x*y/
$$\frac{x^{2} \left(x - y\right)^{2} + x y \left(x - y\right)^{2}}{x^{2} \left(x^{2} + 6 x y + y^{2}\right)}$$
(x^2*(x - y)^2 + x*y*(x - y)^2)/(x^2*(x^2 + y^2 + 6*x*y))
Powers
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2 / 2 \ (y - x) *\x + x*y/ --------------------- 2 / 2 \ x *\(x + y) + 4*x*y/
$$\frac{\left(- x + y\right)^{2} \left(x^{2} + x y\right)}{x^{2} \left(4 x y + \left(x + y\right)^{2}\right)}$$
(y - x)^2*(x^2 + x*y)/(x^2*((x + y)^2 + 4*x*y))
Common denominator
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3 2 2
- y + 2*x*y + 7*y*x
1 - ----------------------
3 2 2
x + x*y + 6*y*x
$$1 - \frac{7 x^{2} y + 2 x y^{2} - y^{3}}{x^{3} + 6 x^{2} y + x y^{2}}$$
1 - (-y^3 + 2*x*y^2 + 7*y*x^2)/(x^3 + x*y^2 + 6*y*x^2)
Combinatorics
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2 (x - y) *(x + y) ------------------- / 2 2 \ x*\x + y + 6*x*y/
$$\frac{\left(x - y\right)^{2} \left(x + y\right)}{x \left(x^{2} + 6 x y + y^{2}\right)}$$
(x - y)^2*(x + y)/(x*(x^2 + y^2 + 6*x*y))
Combining rational expressions
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2 (y - x) *(x + y) -------------------- / 2 \ x*\(x + y) + 4*x*y/
$$\frac{\left(- x + y\right)^{2} \left(x + y\right)}{x \left(4 x y + \left(x + y\right)^{2}\right)}$$
(y - x)^2*(x + y)/(x*((x + y)^2 + 4*x*y))