Mister Exam
Expression simplification/
Least common denominator/
(z^2-2*t+2*t1+exp(t1/t)*(2*t-z^2))*(z^2-2*t+2*t2+exp(t2/t)*(2*t-z^2))
Least common denominator (z^2-2*t+2*t1+exp(t1/t)*(2*t-z^2))*(z^2-2*t+2*t2+exp(t2/t)*(2*t-z^2))
The solution
You have entered
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/ t1 \ / t2 \ | -- | | -- | | 2 t / 2\| | 2 t / 2\| \z - 2*t + 2*t1 + e *\2*t - z //*\z - 2*t + 2*t2 + e *\2*t - z //
$$\left(\left(2 t - z^{2}\right) e^{\frac{t_{1}}{t}} + \left(2 t_{1} + \left(- 2 t + z^{2}\right)\right)\right) \left(\left(2 t - z^{2}\right) e^{\frac{t_{2}}{t}} + \left(2 t_{2} + \left(- 2 t + z^{2}\right)\right)\right)$$
(z^2 - 2*t + 2*t1 + exp(t1/t)*(2*t - z^2))*(z^2 - 2*t + 2*t2 + exp(t2/t)*(2*t - z^2))
General simplification
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/ t1\ / t2\ | --| | --| | 2 / 2 \ t | | 2 / 2 \ t | \z - 2*t + 2*t1 + \- z + 2*t/*e /*\z - 2*t + 2*t2 + \- z + 2*t/*e /
$$\left(- 2 t + 2 t_{1} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{1}}{t}}\right) \left(- 2 t + 2 t_{2} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{2}}{t}}\right)$$
(z^2 - 2*t + 2*t1 + (-z^2 + 2*t)*exp(t1/t))*(z^2 - 2*t + 2*t2 + (-z^2 + 2*t)*exp(t2/t))
Numerical answer
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(z^2 + 2.0*t1 - 2.0*t + (-z^2 + 2.0*t)*exp(t1/t))*(z^2 + 2.0*t2 - 2.0*t + (-z^2 + 2.0*t)*exp(t2/t))
(z^2 + 2.0*t1 - 2.0*t + (-z^2 + 2.0*t)*exp(t1/t))*(z^2 + 2.0*t2 - 2.0*t + (-z^2 + 2.0*t)*exp(t2/t))
Common denominator
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t1 t2 t1 t2 t1 t2 t2 t1 t2 t1 t1 t2 t1 t2 t1 t2
-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
4 2 4 t 4 t 2 2 t 2 t 2 2 4 t t 2 t 2 t t t 2 t 2 t 2 t t 2 t t
z + 4*t - z *e - z *e - 4*t*t1 - 4*t*t2 - 4*t*z - 4*t *e - 4*t *e + 2*t1*z + 2*t2*z + 4*t1*t2 + z *e *e - 2*t1*z *e - 2*t2*z *e + 4*t*t1*e + 4*t*t2*e + 4*t*z *e + 4*t*z *e + 4*t *e *e - 4*t*z *e *e
$$4 t^{2} e^{\frac{t_{1}}{t}} e^{\frac{t_{2}}{t}} - 4 t^{2} e^{\frac{t_{1}}{t}} - 4 t^{2} e^{\frac{t_{2}}{t}} + 4 t^{2} + 4 t t_{1} e^{\frac{t_{2}}{t}} - 4 t t_{1} + 4 t t_{2} e^{\frac{t_{1}}{t}} - 4 t t_{2} - 4 t z^{2} e^{\frac{t_{1}}{t}} e^{\frac{t_{2}}{t}} + 4 t z^{2} e^{\frac{t_{1}}{t}} + 4 t z^{2} e^{\frac{t_{2}}{t}} - 4 t z^{2} + 4 t_{1} t_{2} - 2 t_{1} z^{2} e^{\frac{t_{2}}{t}} + 2 t_{1} z^{2} - 2 t_{2} z^{2} e^{\frac{t_{1}}{t}} + 2 t_{2} z^{2} + z^{4} e^{\frac{t_{1}}{t}} e^{\frac{t_{2}}{t}} - z^{4} e^{\frac{t_{1}}{t}} - z^{4} e^{\frac{t_{2}}{t}} + z^{4}$$
z^4 + 4*t^2 - z^4*exp(t1/t) - z^4*exp(t2/t) - 4*t*t1 - 4*t*t2 - 4*t*z^2 - 4*t^2*exp(t1/t) - 4*t^2*exp(t2/t) + 2*t1*z^2 + 2*t2*z^2 + 4*t1*t2 + z^4*exp(t1/t)*exp(t2/t) - 2*t1*z^2*exp(t2/t) - 2*t2*z^2*exp(t1/t) + 4*t*t1*exp(t2/t) + 4*t*t2*exp(t1/t) + 4*t*z^2*exp(t1/t) + 4*t*z^2*exp(t2/t) + 4*t^2*exp(t1/t)*exp(t2/t) - 4*t*z^2*exp(t1/t)*exp(t2/t)
Combining rational expressions
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/ t1\ / t2\ | --| | --| | 2 / 2 \ t | | 2 / 2 \ t | \z - 2*t + 2*t1 + \- z + 2*t/*e /*\z - 2*t + 2*t2 + \- z + 2*t/*e /
$$\left(- 2 t + 2 t_{1} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{1}}{t}}\right) \left(- 2 t + 2 t_{2} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{2}}{t}}\right)$$
(z^2 - 2*t + 2*t1 + (-z^2 + 2*t)*exp(t1/t))*(z^2 - 2*t + 2*t2 + (-z^2 + 2*t)*exp(t2/t))
Powers
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/ t1\ / t2\ | --| | --| | 2 / 2 \ t | | 2 / 2 \ t | \z - 2*t + 2*t1 + \- z + 2*t/*e /*\z - 2*t + 2*t2 + \- z + 2*t/*e /
$$\left(- 2 t + 2 t_{1} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{1}}{t}}\right) \left(- 2 t + 2 t_{2} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{2}}{t}}\right)$$
(z^2 - 2*t + 2*t1 + (-z^2 + 2*t)*exp(t1/t))*(z^2 - 2*t + 2*t2 + (-z^2 + 2*t)*exp(t2/t))
Trigonometric part
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/ t1\ / t2\ | --| | --| | 2 / 2 \ t | | 2 / 2 \ t | \z - 2*t + 2*t1 + \- z + 2*t/*e /*\z - 2*t + 2*t2 + \- z + 2*t/*e /
$$\left(- 2 t + 2 t_{1} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{1}}{t}}\right) \left(- 2 t + 2 t_{2} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{2}}{t}}\right)$$
/ 2 / 2 \ / /t1\ /t1\\\ / 2 / 2 \ / /t2\ /t2\\\ |z - 2*t + 2*t1 + \- z + 2*t/*|cosh|--| + sinh|--|||*|z - 2*t + 2*t2 + \- z + 2*t/*|cosh|--| + sinh|--||| \ \ \t / \t /// \ \ \t / \t ///
$$\left(- 2 t + 2 t_{1} + z^{2} + \left(2 t - z^{2}\right) \left(\sinh{\left(\frac{t_{1}}{t} \right)} + \cosh{\left(\frac{t_{1}}{t} \right)}\right)\right) \left(- 2 t + 2 t_{2} + z^{2} + \left(2 t - z^{2}\right) \left(\sinh{\left(\frac{t_{2}}{t} \right)} + \cosh{\left(\frac{t_{2}}{t} \right)}\right)\right)$$
(z^2 - 2*t + 2*t1 + (-z^2 + 2*t)*(cosh(t1/t) + sinh(t1/t)))*(z^2 - 2*t + 2*t2 + (-z^2 + 2*t)*(cosh(t2/t) + sinh(t2/t)))
Combinatorics
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/ t1 t1\ / t2 t2\ | -- --| | -- --| | 2 2 t t | | 2 2 t t | \- z - 2*t1 + 2*t + z *e - 2*t*e /*\- z - 2*t2 + 2*t + z *e - 2*t*e /
$$\left(- 2 t e^{\frac{t_{1}}{t}} + 2 t - 2 t_{1} + z^{2} e^{\frac{t_{1}}{t}} - z^{2}\right) \left(- 2 t e^{\frac{t_{2}}{t}} + 2 t - 2 t_{2} + z^{2} e^{\frac{t_{2}}{t}} - z^{2}\right)$$
(-z^2 - 2*t1 + 2*t + z^2*exp(t1/t) - 2*t*exp(t1/t))*(-z^2 - 2*t2 + 2*t + z^2*exp(t2/t) - 2*t*exp(t2/t))
Assemble expression
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/ t1\ / t2\ | --| | --| | 2 / 2 \ t | | 2 / 2 \ t | \z - 2*t + 2*t1 + \- z + 2*t/*e /*\z - 2*t + 2*t2 + \- z + 2*t/*e /
$$\left(- 2 t + 2 t_{1} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{1}}{t}}\right) \left(- 2 t + 2 t_{2} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{2}}{t}}\right)$$
(z^2 - 2*t + 2*t1 + (-z^2 + 2*t)*exp(t1/t))*(z^2 - 2*t + 2*t2 + (-z^2 + 2*t)*exp(t2/t))
Rational denominator
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/ t1\ / t2\ | --| | --| | 2 / 2 \ t | | 2 / 2 \ t | \z - 2*t + 2*t1 + \- z + 2*t/*e /*\z - 2*t + 2*t2 + \- z + 2*t/*e /
$$\left(- 2 t + 2 t_{1} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{1}}{t}}\right) \left(- 2 t + 2 t_{2} + z^{2} + \left(2 t - z^{2}\right) e^{\frac{t_{2}}{t}}\right)$$
(z^2 - 2*t + 2*t1 + (-z^2 + 2*t)*exp(t1/t))*(z^2 - 2*t + 2*t2 + (-z^2 + 2*t)*exp(t2/t))