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Least common denominator k1/(1+k1*(-k2*k3*k3/((t3^2*s^2+2*t3*s*e+1)*(t2*s+1))+1/(t1*s+1)))

An expression to simplify:

The solution

You have entered [src]
                          k1                          
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       /            -k2*k3*k3                   1    \
1 + k1*|---------------------------------- + --------|
       |/  2  2               \              t1*s + 1|
       \\t3 *s  + 2*t3*s*E + 1/*(t2*s + 1)           /
$$\frac{k_{1}}{k_{1} \left(\frac{k_{3} \cdot - k_{2} k_{3}}{\left(s t_{2} + 1\right) \left(\left(s^{2} t_{3}^{2} + e s 2 t_{3}\right) + 1\right)} + \frac{1}{s t_{1} + 1}\right) + 1}$$
k1/(1 + k1*((((-k2)*k3)*k3)/(((t3^2*s^2 + ((2*t3)*s)*E + 1)*(t2*s + 1))) + 1/(t1*s + 1)))
General simplification [src]
                          k1                          
------------------------------------------------------
       /                              2              \
       |   1                     k2*k3               |
1 + k1*|-------- - ----------------------------------|
       |1 + s*t1              /     2   2           \|
       \           (1 + s*t2)*\1 + s *t3  + 2*E*s*t3//
$$\frac{k_{1}}{k_{1} \left(- \frac{k_{2} k_{3}^{2}}{\left(s t_{2} + 1\right) \left(s^{2} t_{3}^{2} + 2 e s t_{3} + 1\right)} + \frac{1}{s t_{1} + 1}\right) + 1}$$
k1/(1 + k1*(1/(1 + s*t1) - k2*k3^2/((1 + s*t2)*(1 + s^2*t3^2 + 2*E*s*t3))))
Expand expression [src]
                          k1                          
------------------------------------------------------
       /                              2              \
       |   1                     k2*k3               |
1 + k1*|-------- - ----------------------------------|
       |t1*s + 1              /  2  2               \|
       \           (t2*s + 1)*\t3 *s  + 2*t3*s*E + 1//
$$\frac{k_{1}}{k_{1} \left(- \frac{k_{2} k_{3}^{2}}{\left(s t_{2} + 1\right) \left(\left(s^{2} t_{3}^{2} + e s 2 t_{3}\right) + 1\right)} + \frac{1}{s t_{1} + 1}\right) + 1}$$
k1/(1 + k1*(1/(t1*s + 1) - k2*k3^2/((t2*s + 1)*(t3^2*s^2 + ((2*t3)*s)*E + 1))))
Numerical answer [src]
k1/(1.0 + k1*(1/(1.0 + s*t1) - k2*k3^2/((1.0 + s*t2)*(1.0 + s^2*t3^2 + 5.43656365691809*s*t3))))
k1/(1.0 + k1*(1/(1.0 + s*t1) - k2*k3^2/((1.0 + s*t2)*(1.0 + s^2*t3^2 + 5.43656365691809*s*t3))))
Combining rational expressions [src]
                              k1*(1 + s*t1)*(1 + s*t2)*(1 + s*t3*(2*E + s*t3))                             
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   /                                          2           \                                                
k1*\(1 + s*t2)*(1 + s*t3*(2*E + s*t3)) - k2*k3 *(1 + s*t1)/ + (1 + s*t1)*(1 + s*t2)*(1 + s*t3*(2*E + s*t3))
$$\frac{k_{1} \left(s t_{1} + 1\right) \left(s t_{2} + 1\right) \left(s t_{3} \left(s t_{3} + 2 e\right) + 1\right)}{k_{1} \left(- k_{2} k_{3}^{2} \left(s t_{1} + 1\right) + \left(s t_{2} + 1\right) \left(s t_{3} \left(s t_{3} + 2 e\right) + 1\right)\right) + \left(s t_{1} + 1\right) \left(s t_{2} + 1\right) \left(s t_{3} \left(s t_{3} + 2 e\right) + 1\right)}$$
k1*(1 + s*t1)*(1 + s*t2)*(1 + s*t3*(2*E + s*t3))/(k1*((1 + s*t2)*(1 + s*t3*(2*E + s*t3)) - k2*k3^2*(1 + s*t1)) + (1 + s*t1)*(1 + s*t2)*(1 + s*t3*(2*E + s*t3)))
Combinatorics [src]
                                                                                                                       /     2   2           \                                                                                              
                                                                                              k1*(1 + s*t1)*(1 + s*t2)*\1 + s *t3  + 2*E*s*t3/                                                                                              
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                        2   2                 2   2          2       3   2       3   2           2                     3   2          4   2                            2              2                2                 2                 3
1 + k1 + s*t1 + s*t2 + s *t3  + k1*s*t2 + k1*s *t3  + t1*t2*s  + t1*s *t3  + t2*s *t3  - k1*k2*k3  + 2*E*s*t3 + k1*t2*s *t3  + t1*t2*s *t3  + 2*E*k1*s*t3 + 2*E*t1*t3*s  + 2*E*t2*t3*s  - k1*k2*s*t1*k3  + 2*E*k1*t2*t3*s  + 2*E*t1*t2*t3*s 
$$\frac{k_{1} \left(s t_{1} + 1\right) \left(s t_{2} + 1\right) \left(s^{2} t_{3}^{2} + 2 e s t_{3} + 1\right)}{- k_{1} k_{2} k_{3}^{2} s t_{1} - k_{1} k_{2} k_{3}^{2} + k_{1} s^{3} t_{2} t_{3}^{2} + 2 e k_{1} s^{2} t_{2} t_{3} + k_{1} s^{2} t_{3}^{2} + k_{1} s t_{2} + 2 e k_{1} s t_{3} + k_{1} + s^{4} t_{1} t_{2} t_{3}^{2} + 2 e s^{3} t_{1} t_{2} t_{3} + s^{3} t_{1} t_{3}^{2} + s^{3} t_{2} t_{3}^{2} + s^{2} t_{1} t_{2} + 2 e s^{2} t_{1} t_{3} + 2 e s^{2} t_{2} t_{3} + s^{2} t_{3}^{2} + s t_{1} + s t_{2} + 2 e s t_{3} + 1}$$
k1*(1 + s*t1)*(1 + s*t2)*(1 + s^2*t3^2 + 2*E*s*t3)/(1 + k1 + s*t1 + s*t2 + s^2*t3^2 + k1*s*t2 + k1*s^2*t3^2 + t1*t2*s^2 + t1*s^3*t3^2 + t2*s^3*t3^2 - k1*k2*k3^2 + 2*E*s*t3 + k1*t2*s^3*t3^2 + t1*t2*s^4*t3^2 + 2*E*k1*s*t3 + 2*E*t1*t3*s^2 + 2*E*t2*t3*s^2 - k1*k2*s*t1*k3^2 + 2*E*k1*t2*t3*s^2 + 2*E*t1*t2*t3*s^3)
Rational denominator [src]
                                                    /                   2   2          2       3   2       3   2                     4   2              2              2                 3\                                                 
                                                 k1*\1 + s*t1 + s*t2 + s *t3  + t1*t2*s  + t1*s *t3  + t2*s *t3  + 2*E*s*t3 + t1*t2*s *t3  + 2*E*t1*t3*s  + 2*E*t2*t3*s  + 2*E*t1*t2*t3*s /                                                 
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                        2   2                 2   2          2       3   2       3   2           2                     3   2          4   2                            2              2                2                 2                 3
1 + k1 + s*t1 + s*t2 + s *t3  + k1*s*t2 + k1*s *t3  + t1*t2*s  + t1*s *t3  + t2*s *t3  - k1*k2*k3  + 2*E*s*t3 + k1*t2*s *t3  + t1*t2*s *t3  + 2*E*k1*s*t3 + 2*E*t1*t3*s  + 2*E*t2*t3*s  - k1*k2*s*t1*k3  + 2*E*k1*t2*t3*s  + 2*E*t1*t2*t3*s 
$$\frac{k_{1} \left(s^{4} t_{1} t_{2} t_{3}^{2} + 2 e s^{3} t_{1} t_{2} t_{3} + s^{3} t_{1} t_{3}^{2} + s^{3} t_{2} t_{3}^{2} + s^{2} t_{1} t_{2} + 2 e s^{2} t_{1} t_{3} + 2 e s^{2} t_{2} t_{3} + s^{2} t_{3}^{2} + s t_{1} + s t_{2} + 2 e s t_{3} + 1\right)}{- k_{1} k_{2} k_{3}^{2} s t_{1} - k_{1} k_{2} k_{3}^{2} + k_{1} s^{3} t_{2} t_{3}^{2} + 2 e k_{1} s^{2} t_{2} t_{3} + k_{1} s^{2} t_{3}^{2} + k_{1} s t_{2} + 2 e k_{1} s t_{3} + k_{1} + s^{4} t_{1} t_{2} t_{3}^{2} + 2 e s^{3} t_{1} t_{2} t_{3} + s^{3} t_{1} t_{3}^{2} + s^{3} t_{2} t_{3}^{2} + s^{2} t_{1} t_{2} + 2 e s^{2} t_{1} t_{3} + 2 e s^{2} t_{2} t_{3} + s^{2} t_{3}^{2} + s t_{1} + s t_{2} + 2 e s t_{3} + 1}$$
k1*(1 + s*t1 + s*t2 + s^2*t3^2 + t1*t2*s^2 + t1*s^3*t3^2 + t2*s^3*t3^2 + 2*E*s*t3 + t1*t2*s^4*t3^2 + 2*E*t1*t3*s^2 + 2*E*t2*t3*s^2 + 2*E*t1*t2*t3*s^3)/(1 + k1 + s*t1 + s*t2 + s^2*t3^2 + k1*s*t2 + k1*s^2*t3^2 + t1*t2*s^2 + t1*s^3*t3^2 + t2*s^3*t3^2 - k1*k2*k3^2 + 2*E*s*t3 + k1*t2*s^3*t3^2 + t1*t2*s^4*t3^2 + 2*E*k1*s*t3 + 2*E*t1*t3*s^2 + 2*E*t2*t3*s^2 - k1*k2*s*t1*k3^2 + 2*E*k1*t2*t3*s^2 + 2*E*t1*t2*t3*s^3)
Powers [src]
                          k1                          
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       /                              2              \
       |   1                     k2*k3               |
1 + k1*|-------- - ----------------------------------|
       |1 + s*t1              /     2   2           \|
       \           (1 + s*t2)*\1 + s *t3  + 2*E*s*t3//
$$\frac{k_{1}}{k_{1} \left(- \frac{k_{2} k_{3}^{2}}{\left(s t_{2} + 1\right) \left(s^{2} t_{3}^{2} + 2 e s t_{3} + 1\right)} + \frac{1}{s t_{1} + 1}\right) + 1}$$
k1/(1 + k1*(1/(1 + s*t1) - k2*k3^2/((1 + s*t2)*(1 + s^2*t3^2 + 2*E*s*t3))))
Common denominator [src]
                                                                       2          2     2  2   2        2   2        2  3   2              2             2   2               2  2                                                                
                                                                     k1  + s*t2*k1  + k1 *s *t3  - k2*k1 *k3  + t2*k1 *s *t3  + 2*E*s*t3*k1  - k2*s*t1*k1 *k3  + 2*E*t2*t3*k1 *s                                                                 
k1 - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                             2   2                 2   2          2       3   2       3   2           2                     3   2          4   2                            2              2                2                 2                 3
     1 + k1 + s*t1 + s*t2 + s *t3  + k1*s*t2 + k1*s *t3  + t1*t2*s  + t1*s *t3  + t2*s *t3  - k1*k2*k3  + 2*E*s*t3 + k1*t2*s *t3  + t1*t2*s *t3  + 2*E*k1*s*t3 + 2*E*t1*t3*s  + 2*E*t2*t3*s  - k1*k2*s*t1*k3  + 2*E*k1*t2*t3*s  + 2*E*t1*t2*t3*s 
$$k_{1} - \frac{- k_{1}^{2} k_{2} k_{3}^{2} s t_{1} - k_{1}^{2} k_{2} k_{3}^{2} + k_{1}^{2} s^{3} t_{2} t_{3}^{2} + 2 e k_{1}^{2} s^{2} t_{2} t_{3} + k_{1}^{2} s^{2} t_{3}^{2} + k_{1}^{2} s t_{2} + 2 e k_{1}^{2} s t_{3} + k_{1}^{2}}{- k_{1} k_{2} k_{3}^{2} s t_{1} - k_{1} k_{2} k_{3}^{2} + k_{1} s^{3} t_{2} t_{3}^{2} + 2 e k_{1} s^{2} t_{2} t_{3} + k_{1} s^{2} t_{3}^{2} + k_{1} s t_{2} + 2 e k_{1} s t_{3} + k_{1} + s^{4} t_{1} t_{2} t_{3}^{2} + 2 e s^{3} t_{1} t_{2} t_{3} + s^{3} t_{1} t_{3}^{2} + s^{3} t_{2} t_{3}^{2} + s^{2} t_{1} t_{2} + 2 e s^{2} t_{1} t_{3} + 2 e s^{2} t_{2} t_{3} + s^{2} t_{3}^{2} + s t_{1} + s t_{2} + 2 e s t_{3} + 1}$$
k1 - (k1^2 + s*t2*k1^2 + k1^2*s^2*t3^2 - k2*k1^2*k3^2 + t2*k1^2*s^3*t3^2 + 2*E*s*t3*k1^2 - k2*s*t1*k1^2*k3^2 + 2*E*t2*t3*k1^2*s^2)/(1 + k1 + s*t1 + s*t2 + s^2*t3^2 + k1*s*t2 + k1*s^2*t3^2 + t1*t2*s^2 + t1*s^3*t3^2 + t2*s^3*t3^2 - k1*k2*k3^2 + 2*E*s*t3 + k1*t2*s^3*t3^2 + t1*t2*s^4*t3^2 + 2*E*k1*s*t3 + 2*E*t1*t3*s^2 + 2*E*t2*t3*s^2 - k1*k2*s*t1*k3^2 + 2*E*k1*t2*t3*s^2 + 2*E*t1*t2*t3*s^3)
Assemble expression [src]
                          k1                          
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       /                              2              \
       |   1                     k2*k3               |
1 + k1*|-------- - ----------------------------------|
       |1 + s*t1              /     2   2           \|
       \           (1 + s*t2)*\1 + s *t3  + 2*E*s*t3//
$$\frac{k_{1}}{k_{1} \left(- \frac{k_{2} k_{3}^{2}}{\left(s t_{2} + 1\right) \left(s^{2} t_{3}^{2} + 2 e s t_{3} + 1\right)} + \frac{1}{s t_{1} + 1}\right) + 1}$$
k1/(1 + k1*(1/(1 + s*t1) - k2*k3^2/((1 + s*t2)*(1 + s^2*t3^2 + 2*E*s*t3))))
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