Mister Exam
Expression simplification/
Least common denominator/
2/(a+2)+(a-2)/(a^2+4*a+4)/(a/(2*a-4)+(a^2+4)/(8-2*a)-2/(2*a+a^2))
Least common denominator 2/(a+2)+(a-2)/(a^2+4*a+4)/(a/(2*a-4)+(a^2+4)/(8-2*a)-2/(2*a+a^2))
The solution
You have entered
[src]
/ a - 2 \
|------------|
| 2 |
2 \a + 4*a + 4/
----- + ----------------------------
a + 2 2
a a + 4 2
------- + ------- - --------
2*a - 4 8 - 2*a 2
2*a + a
$$\frac{\left(a - 2\right) \frac{1}{\left(a^{2} + 4 a\right) + 4}}{\left(\frac{a}{2 a - 4} + \frac{a^{2} + 4}{8 - 2 a}\right) - \frac{2}{a^{2} + 2 a}} + \frac{2}{a + 2}$$
2/(a + 2) + ((a - 2)/(a^2 + 4*a + 4))/(a/(2*a - 4) + (a^2 + 4)/(8 - 2*a) - 2/(2*a + a^2))
General simplification
[src]
/ 5 2 4 3\
2*\32 + a - 24*a - 8*a - 2*a + 10*a /
----------------------------------------
5 6 2 3
64 + a + a - 48*a - 16*a + 16*a
$$\frac{2 \left(a^{5} - 2 a^{4} + 10 a^{3} - 8 a^{2} - 24 a + 32\right)}{a^{6} + a^{5} + 16 a^{3} - 16 a^{2} - 48 a + 64}$$
2*(32 + a^5 - 24*a - 8*a^2 - 2*a^4 + 10*a^3)/(64 + a^5 + a^6 - 48*a - 16*a^2 + 16*a^3)
Fraction decomposition
[src]
-2/(2 + a) + 2*(32 - 48*a - 7*a^3 + 2*a^4 + 26*a^2)/(32 + a^5 - a^4 - 40*a + 2*a^3 + 12*a^2)
$$\frac{2 \left(2 a^{4} - 7 a^{3} + 26 a^{2} - 48 a + 32\right)}{a^{5} - a^{4} + 2 a^{3} + 12 a^{2} - 40 a + 32} - \frac{2}{a + 2}$$
/ 3 4 2\
2 2*\32 - 48*a - 7*a + 2*a + 26*a /
- ----- + -----------------------------------
2 + a 5 4 3 2
32 + a - a - 40*a + 2*a + 12*a
Trigonometric part
[src]
2 -2 + a
----- + ------------------------------------------------
2 + a / 2\
/ 2 \ | 2 a 4 + a |
\4 + a + 4*a/*|- -------- + -------- + -------|
| 2 -4 + 2*a 8 - 2*a|
\ a + 2*a /
$$\frac{a - 2}{\left(a^{2} + 4 a + 4\right) \left(\frac{a}{2 a - 4} - \frac{2}{a^{2} + 2 a} + \frac{a^{2} + 4}{8 - 2 a}\right)} + \frac{2}{a + 2}$$
2/(2 + a) + (-2 + a)/((4 + a^2 + 4*a)*(-2/(a^2 + 2*a) + a/(-4 + 2*a) + (4 + a^2)/(8 - 2*a)))
Rational denominator
[src]
2 3 7 6 5 4
512 - 384*a - 64*a + 4*a + 8*a + 24*a + 96*a + 128*a
---------------------------------------------------------------
/ 2 \ / 4 5 3 2\
(2 + a)*\4 + a + 4*a/*\64 - 80*a - 2*a + 2*a + 4*a + 24*a /
$$\frac{4 a^{7} + 8 a^{6} + 24 a^{5} + 96 a^{4} - 64 a^{3} - 384 a^{2} + 128 a + 512}{\left(a + 2\right) \left(a^{2} + 4 a + 4\right) \left(2 a^{5} - 2 a^{4} + 4 a^{3} + 24 a^{2} - 80 a + 64\right)}$$
(512 - 384*a^2 - 64*a^3 + 4*a^7 + 8*a^6 + 24*a^5 + 96*a^4 + 128*a)/((2 + a)*(4 + a^2 + 4*a)*(64 - 80*a - 2*a^4 + 2*a^5 + 4*a^3 + 24*a^2))
Combining rational expressions
[src]
/ / / / 2\\\ 2 2 \
2*\(4 + a*(4 + a))*\-4*(-2 + a)*(4 - a) + a*(2 + a)*\a*(4 - a) + (-2 + a)*\4 + a /// + a*(-2 + a) *(2 + a) *(4 - a)/
--------------------------------------------------------------------------------------------------------------------
/ / / 2\\\
(2 + a)*(4 + a*(4 + a))*\-4*(-2 + a)*(4 - a) + a*(2 + a)*\a*(4 - a) + (-2 + a)*\4 + a ///
$$\frac{2 \left(a \left(4 - a\right) \left(a - 2\right)^{2} \left(a + 2\right)^{2} + \left(a \left(a + 4\right) + 4\right) \left(a \left(a + 2\right) \left(a \left(4 - a\right) + \left(a - 2\right) \left(a^{2} + 4\right)\right) - 4 \left(4 - a\right) \left(a - 2\right)\right)\right)}{\left(a + 2\right) \left(a \left(a + 4\right) + 4\right) \left(a \left(a + 2\right) \left(a \left(4 - a\right) + \left(a - 2\right) \left(a^{2} + 4\right)\right) - 4 \left(4 - a\right) \left(a - 2\right)\right)}$$
2*((4 + a*(4 + a))*(-4*(-2 + a)*(4 - a) + a*(2 + a)*(a*(4 - a) + (-2 + a)*(4 + a^2))) + a*(-2 + a)^2*(2 + a)^2*(4 - a))/((2 + a)*(4 + a*(4 + a))*(-4*(-2 + a)*(4 - a) + a*(2 + a)*(a*(4 - a) + (-2 + a)*(4 + a^2))))
Powers
[src]
2 -2 + a
----- + ------------------------------------------------
2 + a / 2\
/ 2 \ | 2 a 4 + a |
\4 + a + 4*a/*|- -------- + -------- + -------|
| 2 -4 + 2*a 8 - 2*a|
\ a + 2*a /
$$\frac{a - 2}{\left(a^{2} + 4 a + 4\right) \left(\frac{a}{2 a - 4} - \frac{2}{a^{2} + 2 a} + \frac{a^{2} + 4}{8 - 2 a}\right)} + \frac{2}{a + 2}$$
2/(2 + a) + (-2 + a)/((4 + a^2 + 4*a)*(-2/(a^2 + 2*a) + a/(-4 + 2*a) + (4 + a^2)/(8 - 2*a)))
Combinatorics
[src]
/ 5 2 4 3\
2*\32 + a - 24*a - 8*a - 2*a + 10*a /
--------------------------------------------
/ 5 4 3 2\
(2 + a)*\32 + a - a - 40*a + 2*a + 12*a /
$$\frac{2 \left(a^{5} - 2 a^{4} + 10 a^{3} - 8 a^{2} - 24 a + 32\right)}{\left(a + 2\right) \left(a^{5} - a^{4} + 2 a^{3} + 12 a^{2} - 40 a + 32\right)}$$
2*(32 + a^5 - 24*a - 8*a^2 - 2*a^4 + 10*a^3)/((2 + a)*(32 + a^5 - a^4 - 40*a + 2*a^3 + 12*a^2))
Assemble expression
[src]
2 -2 + a
----- + ------------------------------------------------
2 + a / 2\
/ 2 \ | 2 a 4 + a |
\4 + a + 4*a/*|- -------- + -------- + -------|
| 2 -4 + 2*a 8 - 2*a|
\ a + 2*a /
$$\frac{a - 2}{\left(a^{2} + 4 a + 4\right) \left(\frac{a}{2 a - 4} - \frac{2}{a^{2} + 2 a} + \frac{a^{2} + 4}{8 - 2 a}\right)} + \frac{2}{a + 2}$$
2/(2 + a) + (-2 + a)/((4 + a^2 + 4*a)*(-2/(a^2 + 2*a) + a/(-4 + 2*a) + (4 + a^2)/(8 - 2*a)))
Numerical answer
[src]
2.0/(2.0 + a) + (-2.0 + a)/((4.0 + a^2 + 4.0*a)*(-2.0/(a^2 + 2.0*a) + a/(-4.0 + 2.0*a) + (4.0 + a^2)/(8.0 - 2.0*a)))
2.0/(2.0 + a) + (-2.0 + a)/((4.0 + a^2 + 4.0*a)*(-2.0/(a^2 + 2.0*a) + a/(-4.0 + 2.0*a) + (4.0 + a^2)/(8.0 - 2.0*a)))
Common denominator
[src]
2 4 5 3
64 - 48*a - 16*a - 4*a + 2*a + 20*a
---------------------------------------
5 6 2 3
64 + a + a - 48*a - 16*a + 16*a
$$\frac{2 a^{5} - 4 a^{4} + 20 a^{3} - 16 a^{2} - 48 a + 64}{a^{6} + a^{5} + 16 a^{3} - 16 a^{2} - 48 a + 64}$$
(64 - 48*a - 16*a^2 - 4*a^4 + 2*a^5 + 20*a^3)/(64 + a^5 + a^6 - 48*a - 16*a^2 + 16*a^3)