Mister Exam
Least common denominator ((y/x-y)+(x/x+y))*((x^2/y^2)+(y^2/x^2)-2)
The solution
You have entered
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/ 2 2 \
/y x \ |x y |
|- - y + - + y|*|-- + -- - 2|
\x x / | 2 2 |
\y x /
$$\left(\left(- y + \frac{y}{x}\right) + \left(y + \frac{x}{x}\right)\right) \left(\left(\frac{x^{2}}{y^{2}} + \frac{y^{2}}{x^{2}}\right) - 2\right)$$
(y/x - y + x/x + y)*(x^2/y^2 + y^2/x^2 - 2)
General simplification
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/ 4 4 2 2\
(x + y)*\x + y - 2*x *y /
---------------------------
3 2
x *y
$$\frac{\left(x + y\right) \left(x^{4} - 2 x^{2} y^{2} + y^{4}\right)}{x^{3} y^{2}}$$
(x + y)*(x^4 + y^4 - 2*x^2*y^2)/(x^3*y^2)
Combinatorics
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3 2
(x + y) *(x - y)
-----------------
3 2
x *y
$$\frac{\left(x - y\right)^{2} \left(x + y\right)^{3}}{x^{3} y^{2}}$$
(x + y)^3*(x - y)^2/(x^3*y^2)
Common denominator
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5 5 4 4 2 3
x + y + x*y + y*x - 2*x *y
-2 + -------------------------------
3 2
x *y
$$-2 + \frac{x^{5} + x^{4} y - 2 x^{2} y^{3} + x y^{4} + y^{5}}{x^{3} y^{2}}$$
-2 + (x^5 + y^5 + x*y^4 + y*x^4 - 2*x^2*y^3)/(x^3*y^2)
Rational denominator
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/ 4 4 2 2\
(x + y)*\x + y - 2*x *y /
---------------------------
3 2
x *y
$$\frac{\left(x + y\right) \left(x^{4} - 2 x^{2} y^{2} + y^{4}\right)}{x^{3} y^{2}}$$
(x + y)*(x^4 + y^4 - 2*x^2*y^2)/(x^3*y^2)
Trigonometric part
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/ 2 2\
/ y\ | y x |
|1 + -|*|-2 + -- + --|
\ x/ | 2 2|
\ x y /
$$\left(1 + \frac{y}{x}\right) \left(\frac{x^{2}}{y^{2}} - 2 + \frac{y^{2}}{x^{2}}\right)$$
(1 + y/x)*(-2 + y^2/x^2 + x^2/y^2)
Assemble expression
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/ 2 2\
/ y\ | y x |
|1 + -|*|-2 + -- + --|
\ x/ | 2 2|
\ x y /
$$\left(1 + \frac{y}{x}\right) \left(\frac{x^{2}}{y^{2}} - 2 + \frac{y^{2}}{x^{2}}\right)$$
(1 + y/x)*(-2 + y^2/x^2 + x^2/y^2)
Combining rational expressions
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/ 4 4 2 2\
(x*(1 + y) + y*(1 - x))*\x + y - 2*x *y /
-------------------------------------------
3 2
x *y
$$\frac{\left(x \left(y + 1\right) + y \left(1 - x\right)\right) \left(x^{4} - 2 x^{2} y^{2} + y^{4}\right)}{x^{3} y^{2}}$$
(x*(1 + y) + y*(1 - x))*(x^4 + y^4 - 2*x^2*y^2)/(x^3*y^2)
Powers
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/ 2 2\
/ y\ | y x |
|1 + -|*|-2 + -- + --|
\ x/ | 2 2|
\ x y /
$$\left(1 + \frac{y}{x}\right) \left(\frac{x^{2}}{y^{2}} - 2 + \frac{y^{2}}{x^{2}}\right)$$
(1 + y/x)*(-2 + y^2/x^2 + x^2/y^2)