Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sqrt(x-2*sqrt(x+3)+4)/(1-sqrt(x+3))*(sqrt(x)-sqrt(x+3))-(1/(sqrt(x)+sqrt(x-3)))
¿Cómo vas a descomponer esta sqrt(x-2*sqrt(x+3)+4)/(1-sqrt(x+3))*(sqrt(x)-sqrt(x+3))-(1/(sqrt(x)+sqrt(x-3))) expresión en fracciones?
Solución
Ha introducido
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_____________________
/ _______
\/ x - 2*\/ x + 3 + 4 / ___ _______\ 1
------------------------*\\/ x - \/ x + 3 / - -----------------
_______ ___ _______
1 - \/ x + 3 \/ x + \/ x - 3
$$\frac{\sqrt{\left(x - 2 \sqrt{x + 3}\right) + 4}}{1 - \sqrt{x + 3}} \left(\sqrt{x} - \sqrt{x + 3}\right) - \frac{1}{\sqrt{x} + \sqrt{x - 3}}$$
(sqrt(x - 2*sqrt(x + 3) + 4)/(1 - sqrt(x + 3)))*(sqrt(x) - sqrt(x + 3)) - 1/(sqrt(x) + sqrt(x - 3))
Simplificación general
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_____________________
_______ / _______ / ___ ________\ / ___ _______\
1 - \/ 3 + x - \/ 4 + x - 2*\/ 3 + x *\\/ x + \/ -3 + x /*\\/ x - \/ 3 + x /
---------------------------------------------------------------------------------
/ _______\ / ___ ________\
\-1 + \/ 3 + x /*\\/ x + \/ -3 + x /
$$\frac{- \left(\sqrt{x} + \sqrt{x - 3}\right) \left(\sqrt{x} - \sqrt{x + 3}\right) \sqrt{x - 2 \sqrt{x + 3} + 4} - \sqrt{x + 3} + 1}{\left(\sqrt{x} + \sqrt{x - 3}\right) \left(\sqrt{x + 3} - 1\right)}$$
(1 - sqrt(3 + x) - sqrt(4 + x - 2*sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x))*(sqrt(x) - sqrt(3 + x)))/((-1 + sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x)))
Respuesta numérica
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-1/(x^0.5 + 1.73205080756888*(-1 + 0.333333333333333*x)^0.5) + 2.0*(1 + 0.25*x - 0.866025403784439*(1 + 0.333333333333333*x)^0.5)^0.5*(x^0.5 - 1.73205080756888*(1 + 0.333333333333333*x)^0.5)/(1.0 - 1.73205080756888*(1 + 0.333333333333333*x)^0.5)
-1/(x^0.5 + 1.73205080756888*(-1 + 0.333333333333333*x)^0.5) + 2.0*(1 + 0.25*x - 0.866025403784439*(1 + 0.333333333333333*x)^0.5)^0.5*(x^0.5 - 1.73205080756888*(1 + 0.333333333333333*x)^0.5)/(1.0 - 1.73205080756888*(1 + 0.333333333333333*x)^0.5)
Unión de expresiones racionales
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_____________________
_______ / _______ / ___ ________\ / ___ _______\
-1 + \/ 3 + x + \/ 4 + x - 2*\/ 3 + x *\\/ x + \/ -3 + x /*\\/ x - \/ 3 + x /
----------------------------------------------------------------------------------
/ _______\ / ___ ________\
\1 - \/ 3 + x /*\\/ x + \/ -3 + x /
$$\frac{\left(\sqrt{x} + \sqrt{x - 3}\right) \left(\sqrt{x} - \sqrt{x + 3}\right) \sqrt{x - 2 \sqrt{x + 3} + 4} + \sqrt{x + 3} - 1}{\left(1 - \sqrt{x + 3}\right) \left(\sqrt{x} + \sqrt{x - 3}\right)}$$
(-1 + sqrt(3 + x) + sqrt(4 + x - 2*sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x))*(sqrt(x) - sqrt(3 + x)))/((1 - sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x)))
Denominador racional
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_____________________ _____________________ _____________________ _____________________ _____________________
3/2 ___ ________ / _______ ________ ___ / _______ / _______ _______ / _______ ___ _______ / _______
- x - 2*\/ x + 2*\/ -3 + x + 9*\/ 4 + x - 2*\/ 3 + x + x*\/ -3 + x - 3*\/ x *\/ 4 + x - 2*\/ 3 + x + 3*x*\/ 4 + x - 2*\/ 3 + x + 3*\/ 3 + x *\/ 4 + x - 2*\/ 3 + x - 3*\/ x *\/ 3 + x *\/ 4 + x - 2*\/ 3 + x
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
6 + 3*x
$$\frac{- x^{\frac{3}{2}} - 3 \sqrt{x} \sqrt{x + 3} \sqrt{x - 2 \sqrt{x + 3} + 4} - 3 \sqrt{x} \sqrt{x - 2 \sqrt{x + 3} + 4} - 2 \sqrt{x} + x \sqrt{x - 3} + 3 x \sqrt{x - 2 \sqrt{x + 3} + 4} + 2 \sqrt{x - 3} + 3 \sqrt{x + 3} \sqrt{x - 2 \sqrt{x + 3} + 4} + 9 \sqrt{x - 2 \sqrt{x + 3} + 4}}{3 x + 6}$$
(-x^(3/2) - 2*sqrt(x) + 2*sqrt(-3 + x) + 9*sqrt(4 + x - 2*sqrt(3 + x)) + x*sqrt(-3 + x) - 3*sqrt(x)*sqrt(4 + x - 2*sqrt(3 + x)) + 3*x*sqrt(4 + x - 2*sqrt(3 + x)) + 3*sqrt(3 + x)*sqrt(4 + x - 2*sqrt(3 + x)) - 3*sqrt(x)*sqrt(3 + x)*sqrt(4 + x - 2*sqrt(3 + x)))/(6 + 3*x)
Parte trigonométrica
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_____________________
/ _______ / ___ _______\
1 \/ 4 + x - 2*\/ 3 + x *\\/ x - \/ 3 + x /
- ------------------ + --------------------------------------------
___ ________ _______
\/ x + \/ -3 + x 1 - \/ 3 + x
$$- \frac{1}{\sqrt{x} + \sqrt{x - 3}} + \frac{\left(\sqrt{x} - \sqrt{x + 3}\right) \sqrt{x - 2 \sqrt{x + 3} + 4}}{1 - \sqrt{x + 3}}$$
-1/(sqrt(x) + sqrt(-3 + x)) + sqrt(4 + x - 2*sqrt(3 + x))*(sqrt(x) - sqrt(3 + x))/(1 - sqrt(3 + x))
Denominador común
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_____________________ _____________________ _____________________ _____________________
_____________________ _______ / _______ ___ / _______ ________ / _______ ___ ________ / _______
/ _______ -1 + \/ 3 + x + x*\/ 4 + x - 2*\/ 3 + x - \/ x *\/ 4 + x - 2*\/ 3 + x - \/ -3 + x *\/ 4 + x - 2*\/ 3 + x + \/ x *\/ -3 + x *\/ 4 + x - 2*\/ 3 + x
\/ 4 + x - 2*\/ 3 + x - --------------------------------------------------------------------------------------------------------------------------------------------------------------
___ ________ ___ _______ ________ _______
- \/ x - \/ -3 + x + \/ x *\/ 3 + x + \/ -3 + x *\/ 3 + x
$$\sqrt{x - 2 \sqrt{x + 3} + 4} - \frac{\sqrt{x} \sqrt{x - 3} \sqrt{x - 2 \sqrt{x + 3} + 4} - \sqrt{x} \sqrt{x - 2 \sqrt{x + 3} + 4} + x \sqrt{x - 2 \sqrt{x + 3} + 4} - \sqrt{x - 3} \sqrt{x - 2 \sqrt{x + 3} + 4} + \sqrt{x + 3} - 1}{\sqrt{x} \sqrt{x + 3} - \sqrt{x} + \sqrt{x - 3} \sqrt{x + 3} - \sqrt{x - 3}}$$
sqrt(4 + x - 2*sqrt(3 + x)) - (-1 + sqrt(3 + x) + x*sqrt(4 + x - 2*sqrt(3 + x)) - sqrt(x)*sqrt(4 + x - 2*sqrt(3 + x)) - sqrt(-3 + x)*sqrt(4 + x - 2*sqrt(3 + x)) + sqrt(x)*sqrt(-3 + x)*sqrt(4 + x - 2*sqrt(3 + x)))/(-sqrt(x) - sqrt(-3 + x) + sqrt(x)*sqrt(3 + x) + sqrt(-3 + x)*sqrt(3 + x))
Combinatoria
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/ _____________________ _____________________ _____________________ _____________________\
| _______ / _______ ___ ________ / _______ ___ _______ / _______ ________ _______ / _______ |
-\-1 + \/ 3 + x + x*\/ 4 + x - 2*\/ 3 + x + \/ x *\/ -3 + x *\/ 4 + x - 2*\/ 3 + x - \/ x *\/ 3 + x *\/ 4 + x - 2*\/ 3 + x - \/ -3 + x *\/ 3 + x *\/ 4 + x - 2*\/ 3 + x /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ _______\ / ___ ________\
\-1 + \/ 3 + x /*\\/ x + \/ -3 + x /
$$- \frac{\sqrt{x} \sqrt{x - 3} \sqrt{x - 2 \sqrt{x + 3} + 4} - \sqrt{x} \sqrt{x + 3} \sqrt{x - 2 \sqrt{x + 3} + 4} + x \sqrt{x - 2 \sqrt{x + 3} + 4} - \sqrt{x - 3} \sqrt{x + 3} \sqrt{x - 2 \sqrt{x + 3} + 4} + \sqrt{x + 3} - 1}{\left(\sqrt{x} + \sqrt{x - 3}\right) \left(\sqrt{x + 3} - 1\right)}$$
-(-1 + sqrt(3 + x) + x*sqrt(4 + x - 2*sqrt(3 + x)) + sqrt(x)*sqrt(-3 + x)*sqrt(4 + x - 2*sqrt(3 + x)) - sqrt(x)*sqrt(3 + x)*sqrt(4 + x - 2*sqrt(3 + x)) - sqrt(-3 + x)*sqrt(3 + x)*sqrt(4 + x - 2*sqrt(3 + x)))/((-1 + sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x)))
Potencias
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_____________________
/ _______ / ___ _______\
1 \/ 4 + x - 2*\/ 3 + x *\\/ x - \/ 3 + x /
- ------------------ + --------------------------------------------
___ ________ _______
\/ x + \/ -3 + x 1 - \/ 3 + x
$$- \frac{1}{\sqrt{x} + \sqrt{x - 3}} + \frac{\left(\sqrt{x} - \sqrt{x + 3}\right) \sqrt{x - 2 \sqrt{x + 3} + 4}}{1 - \sqrt{x + 3}}$$
-1/(sqrt(x) + sqrt(-3 + x)) + sqrt(4 + x - 2*sqrt(3 + x))*(sqrt(x) - sqrt(3 + x))/(1 - sqrt(3 + x))
Compilar la expresión
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_____________________
/ _______ / ___ _______\
1 \/ 4 + x - 2*\/ 3 + x *\\/ x - \/ 3 + x /
- ------------------ + --------------------------------------------
___ ________ _______
\/ x + \/ -3 + x 1 - \/ 3 + x
$$- \frac{1}{\sqrt{x} + \sqrt{x - 3}} + \frac{\left(\sqrt{x} - \sqrt{x + 3}\right) \sqrt{x - 2 \sqrt{x + 3} + 4}}{1 - \sqrt{x + 3}}$$
-1/(sqrt(x) + sqrt(-3 + x)) + sqrt(4 + x - 2*sqrt(3 + x))*(sqrt(x) - sqrt(3 + x))/(1 - sqrt(3 + x))