Mister Exam
Expression simplification/
Least common denominator/
(y/(x*y-x^2)+x/(x*y-y^2))/(x^2+2*x*y+y^2/(1/x+1/y))
Least common denominator (y/(x*y-x^2)+x/(x*y-y^2))/(x^2+2*x*y+y^2/(1/x+1/y))
The solution
You have entered
[src]
y x
-------- + --------
2 2
x*y - x x*y - y
-------------------
2
2 y
x + 2*x*y + -----
1 1
- + -
x y
$$\frac{\frac{x}{x y - y^{2}} + \frac{y}{- x^{2} + x y}}{\frac{y^{2}}{\frac{1}{y} + \frac{1}{x}} + \left(x^{2} + 2 x y\right)}$$
(y/(x*y - x^2) + x/(x*y - y^2))/(x^2 + (2*x)*y + y^2/(1/x + 1/y))
General simplification
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/ 2 2\
(x + y)*\x - y /
-------------------------------------
2 / 3 \
x *y*(x - y)*\y + (x + y)*(x + 2*y)/
$$\frac{\left(x + y\right) \left(x^{2} - y^{2}\right)}{x^{2} y \left(x - y\right) \left(y^{3} + \left(x + y\right) \left(x + 2 y\right)\right)}$$
(x + y)*(x^2 - y^2)/(x^2*y*(x - y)*(y^3 + (x + y)*(x + 2*y)))
Rational denominator
[src]
4 4 2 2
x + y - 2*x *y
-----------------------------------------------------
/ 2 \ / 2 \ / 3 3 2 2\
\x - x*y/*\- y + x*y/*\x + x*y + 2*x*y + 3*y*x /
$$\frac{x^{4} - 2 x^{2} y^{2} + y^{4}}{\left(x^{2} - x y\right) \left(x y - y^{2}\right) \left(x^{3} + 3 x^{2} y + x y^{3} + 2 x y^{2}\right)}$$
(x^4 + y^4 - 2*x^2*y^2)/((x^2 - x*y)*(-y^2 + x*y)*(x^3 + x*y^3 + 2*x*y^2 + 3*y*x^2))
Powers
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x y
---------- + ----------
2 2
- y + x*y - x + x*y
-----------------------
2
2 y
x + ----- + 2*x*y
1 1
- + -
x y
$$\frac{\frac{x}{x y - y^{2}} + \frac{y}{- x^{2} + x y}}{x^{2} + 2 x y + \frac{y^{2}}{\frac{1}{y} + \frac{1}{x}}}$$
(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2/(1/x + 1/y) + 2*x*y)
Combining rational expressions
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/ 2 2 \
(x + y)*\x *(y - x) + y *(x - y)/
---------------------------------------------
2 / 3 \
x *y*(x - y)*(y - x)*\y + (x + y)*(x + 2*y)/
$$\frac{\left(x + y\right) \left(x^{2} \left(- x + y\right) + y^{2} \left(x - y\right)\right)}{x^{2} y \left(- x + y\right) \left(x - y\right) \left(y^{3} + \left(x + y\right) \left(x + 2 y\right)\right)}$$
(x + y)*(x^2*(y - x) + y^2*(x - y))/(x^2*y*(x - y)*(y - x)*(y^3 + (x + y)*(x + 2*y)))
Common denominator
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2 2
x + y + 2*x*y
--------------------------------
4 2 4 2 3 3 2
y*x + x *y + 2*x *y + 3*x *y
$$\frac{x^{2} + 2 x y + y^{2}}{x^{4} y + 3 x^{3} y^{2} + x^{2} y^{4} + 2 x^{2} y^{3}}$$
(x^2 + y^2 + 2*x*y)/(y*x^4 + x^2*y^4 + 2*x^2*y^3 + 3*x^3*y^2)
Combinatorics
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2
(x + y)
-----------------------------
2 / 2 3 2 \
x *y*\x + y + 2*y + 3*x*y/
$$\frac{\left(x + y\right)^{2}}{x^{2} y \left(x^{2} + 3 x y + y^{3} + 2 y^{2}\right)}$$
(x + y)^2/(x^2*y*(x^2 + y^3 + 2*y^2 + 3*x*y))
Assemble expression
[src]
x y
---------- + ----------
2 2
- y + x*y - x + x*y
-----------------------
2
2 y
x + ----- + 2*x*y
1 1
- + -
x y
$$\frac{\frac{x}{x y - y^{2}} + \frac{y}{- x^{2} + x y}}{x^{2} + 2 x y + \frac{y^{2}}{\frac{1}{y} + \frac{1}{x}}}$$
(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2/(1/x + 1/y) + 2*x*y)
Numerical answer
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(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2/(1/x + 1/y) + 2.0*x*y)
(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2/(1/x + 1/y) + 2.0*x*y)
Trigonometric part
[src]
x y
---------- + ----------
2 2
- y + x*y - x + x*y
-----------------------
2
2 y
x + ----- + 2*x*y
1 1
- + -
x y
$$\frac{\frac{x}{x y - y^{2}} + \frac{y}{- x^{2} + x y}}{x^{2} + 2 x y + \frac{y^{2}}{\frac{1}{y} + \frac{1}{x}}}$$
(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2/(1/x + 1/y) + 2*x*y)