Mister Exam
Least common denominator (x1/3-y1/3)/(x+y)+1/(x2/3-x1/3*y1/3+y2/3)
The solution
You have entered
[src]
x1 y1
-- - --
3 3 1
------- + ---------------
x + y x1
--*y1
x2 3 y2
-- - ----- + --
3 3 3
$$\frac{1}{\frac{y_{2}}{3} + \left(\frac{x_{2}}{3} - \frac{y_{1} \frac{x_{1}}{3}}{3}\right)} + \frac{\frac{x_{1}}{3} - \frac{y_{1}}{3}}{x + y}$$
(x1/3 - y1/3)/(x + y) + 1/(x2/3 - (x1/3)*y1/3 + y2/3)
General simplification
[src]
27*x + 27*y + (x1 - y1)*(3*x2 + 3*y2 - x1*y1)
---------------------------------------------
3*(x + y)*(3*x2 + 3*y2 - x1*y1)
$$\frac{27 x + 27 y + \left(x_{1} - y_{1}\right) \left(- x_{1} y_{1} + 3 x_{2} + 3 y_{2}\right)}{3 \left(x + y\right) \left(- x_{1} y_{1} + 3 x_{2} + 3 y_{2}\right)}$$
(27*x + 27*y + (x1 - y1)*(3*x2 + 3*y2 - x1*y1))/(3*(x + y)*(3*x2 + 3*y2 - x1*y1))
Common denominator
[src]
/ 2 2 \
-\27*x + 27*y + x1*y1 - y1*x1 - 3*x2*y1 - 3*y1*y2 + 3*x1*x2 + 3*x1*y2/
-------------------------------------------------------------------------
-9*x*x2 - 9*x*y2 - 9*x2*y - 9*y*y2 + 3*x*x1*y1 + 3*x1*y*y1
$$- \frac{27 x - x_{1}^{2} y_{1} + 3 x_{1} x_{2} + x_{1} y_{1}^{2} + 3 x_{1} y_{2} - 3 x_{2} y_{1} + 27 y - 3 y_{1} y_{2}}{3 x x_{1} y_{1} - 9 x x_{2} - 9 x y_{2} + 3 x_{1} y y_{1} - 9 x_{2} y - 9 y y_{2}}$$
-(27*x + 27*y + x1*y1^2 - y1*x1^2 - 3*x2*y1 - 3*y1*y2 + 3*x1*x2 + 3*x1*y2)/(-9*x*x2 - 9*x*y2 - 9*x2*y - 9*y*y2 + 3*x*x1*y1 + 3*x1*y*y1)
Trigonometric part
[src]
y1 x1
- -- + --
1 3 3
--------------- + ---------
x2 y2 x1*y1 x + y
-- + -- - -----
3 3 9
$$\frac{1}{- \frac{x_{1} y_{1}}{9} + \frac{x_{2}}{3} + \frac{y_{2}}{3}} + \frac{\frac{x_{1}}{3} - \frac{y_{1}}{3}}{x + y}$$
1/(x2/3 + y2/3 - x1*y1/9) + (-y1/3 + x1/3)/(x + y)
Numerical answer
[src]
1/(0.333333333333333*x2 + 0.333333333333333*y2 - 0.111111111111111*x1*y1) + (0.333333333333333*x1 - 0.333333333333333*y1)/(x + y)
1/(0.333333333333333*x2 + 0.333333333333333*y2 - 0.111111111111111*x1*y1) + (0.333333333333333*x1 - 0.333333333333333*y1)/(x + y)
Powers
[src]
y1 x1
- -- + --
1 3 3
--------------- + ---------
x2 y2 x1*y1 x + y
-- + -- - -----
3 3 9
$$\frac{1}{- \frac{x_{1} y_{1}}{9} + \frac{x_{2}}{3} + \frac{y_{2}}{3}} + \frac{\frac{x_{1}}{3} - \frac{y_{1}}{3}}{x + y}$$
1/(x2/3 + y2/3 - x1*y1/9) + (-y1/3 + x1/3)/(x + y)
Rational denominator
[src]
243*x + 243*y + (x1 - y1)*(27*x2 + 27*y2 - 9*x1*y1)
---------------------------------------------------
(3*x + 3*y)*(27*x2 + 27*y2 - 9*x1*y1)
$$\frac{243 x + 243 y + \left(x_{1} - y_{1}\right) \left(- 9 x_{1} y_{1} + 27 x_{2} + 27 y_{2}\right)}{\left(3 x + 3 y\right) \left(- 9 x_{1} y_{1} + 27 x_{2} + 27 y_{2}\right)}$$
(243*x + 243*y + (x1 - y1)*(27*x2 + 27*y2 - 9*x1*y1))/((3*x + 3*y)*(27*x2 + 27*y2 - 9*x1*y1))
Combinatorics
[src]
/ 2 2 \
-\27*x + 27*y + x1*y1 - y1*x1 - 3*x2*y1 - 3*y1*y2 + 3*x1*x2 + 3*x1*y2/
-------------------------------------------------------------------------
3*(x + y)*(-3*x2 - 3*y2 + x1*y1)
$$- \frac{27 x - x_{1}^{2} y_{1} + 3 x_{1} x_{2} + x_{1} y_{1}^{2} + 3 x_{1} y_{2} - 3 x_{2} y_{1} + 27 y - 3 y_{1} y_{2}}{3 \left(x + y\right) \left(x_{1} y_{1} - 3 x_{2} - 3 y_{2}\right)}$$
-(27*x + 27*y + x1*y1^2 - y1*x1^2 - 3*x2*y1 - 3*y1*y2 + 3*x1*x2 + 3*x1*y2)/(3*(x + y)*(-3*x2 - 3*y2 + x1*y1))
Combining rational expressions
[src]
27*x + 27*y + (x1 - y1)*(3*x2 + 3*y2 - x1*y1)
---------------------------------------------
3*(x + y)*(3*x2 + 3*y2 - x1*y1)
$$\frac{27 x + 27 y + \left(x_{1} - y_{1}\right) \left(- x_{1} y_{1} + 3 x_{2} + 3 y_{2}\right)}{3 \left(x + y\right) \left(- x_{1} y_{1} + 3 x_{2} + 3 y_{2}\right)}$$
(27*x + 27*y + (x1 - y1)*(3*x2 + 3*y2 - x1*y1))/(3*(x + y)*(3*x2 + 3*y2 - x1*y1))
Assemble expression
[src]
y1 x1
- -- + --
1 3 3
--------------- + ---------
x2 y2 x1*y1 x + y
-- + -- - -----
3 3 9
$$\frac{1}{- \frac{x_{1} y_{1}}{9} + \frac{x_{2}}{3} + \frac{y_{2}}{3}} + \frac{\frac{x_{1}}{3} - \frac{y_{1}}{3}}{x + y}$$
1/(x2/3 + y2/3 - x1*y1/9) + (-y1/3 + x1/3)/(x + y)