Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sqrt(sqrt(2)/a+a/sqrt(2)+2)-(a^2*sqrt(sqrt(2))-2*sqrt(a))/(a*sqrt(2*a)-sqrt(sqrt(8*a^4)))
¿Cómo vas a descomponer esta sqrt(sqrt(2)/a+a/sqrt(2)+2)-(a^2*sqrt(sqrt(2))-2*sqrt(a))/(a*sqrt(2*a)-sqrt(sqrt(8*a^4))) expresión en fracciones?
Solución
Ha introducido
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___________________ _______
/ ___ 2 / ___ ___
/ \/ 2 a a *\/ \/ 2 - 2*\/ a
/ ----- + ----- + 2 - ---------------------------
/ a ___ ___________
\/ \/ 2 / ______
_____ / / 4
a*\/ 2*a - \/ \/ 8*a
$$- \frac{- 2 \sqrt{a} + a^{2} \sqrt{\sqrt{2}}}{a \sqrt{2 a} - \sqrt{\sqrt{8 a^{4}}}} + \sqrt{\left(\frac{a}{\sqrt{2}} + \frac{\sqrt{2}}{a}\right) + 2}$$
sqrt(sqrt(2)/a + a/sqrt(2) + 2) - (a^2*sqrt(sqrt(2)) - 2*sqrt(a))/(a*sqrt(2*a) - sqrt(sqrt(8*a^4)))
Simplificación general
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__________________________
/ ___ ___ 2 / ____\
___ / 2*\/ 2 + 4*a + \/ 2 *a | ___ 3/2 3/4 4 / 4 |
\/ 2 * / ------------------------ *\\/ 2 *a - 2 *\/ a /
___ 4 ___ 2 \/ a
2*\/ a - \/ 2 *a + -----------------------------------------------------------------
2
--------------------------------------------------------------------------------------
____
___ 3/2 3/4 4 / 4
\/ 2 *a - 2 *\/ a
$$\frac{2 \sqrt{a} - \sqrt[4]{2} a^{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2} a^{2} + 4 a + 2 \sqrt{2}}{a}} \left(\sqrt{2} a^{\frac{3}{2}} - 2^{\frac{3}{4}} \sqrt[4]{a^{4}}\right)}{2}}{\sqrt{2} a^{\frac{3}{2}} - 2^{\frac{3}{4}} \sqrt[4]{a^{4}}}$$
(2*sqrt(a) - 2^(1/4)*a^2 + sqrt(2)*sqrt((2*sqrt(2) + 4*a + sqrt(2)*a^2)/a)*(sqrt(2)*a^(3/2) - 2^(3/4)*(a^4)^(1/4))/2)/(sqrt(2)*a^(3/2) - 2^(3/4)*(a^4)^(1/4))
Respuesta numérica
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1.4142135623731*(1 + 0.353553390593274*a + 0.707106781186548/a)^0.5 - (1.18920711500272*a^2 - 2.0*a^0.5)/(1.4142135623731*a^1.5 - 1.68179283050743*(a^4)^0.25)
1.4142135623731*(1 + 0.353553390593274*a + 0.707106781186548/a)^0.5 - (1.18920711500272*a^2 - 2.0*a^0.5)/(1.4142135623731*a^1.5 - 1.68179283050743*(a^4)^0.25)
Compilar la expresión
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_____________________
/ ___ ___ ___ 4 ___ 2
/ \/ 2 a*\/ 2 - 2*\/ a + \/ 2 *a
/ 2 + ----- + ------- - -------------------------
\/ a 2 ____
___ 3/2 3/4 4 / 4
\/ 2 *a - 2 *\/ a
$$- \frac{- 2 \sqrt{a} + \sqrt[4]{2} a^{2}}{\sqrt{2} a^{\frac{3}{2}} - 2^{\frac{3}{4}} \sqrt[4]{a^{4}}} + \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}}$$
sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) - (-2*sqrt(a) + 2^(1/4)*a^2)/(sqrt(2)*a^(3/2) - 2^(3/4)*(a^4)^(1/4))
Denominador común
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_____________________
/ ___ ___ ___ 4 ___ 2
/ \/ 2 a*\/ 2 - 2*\/ a + \/ 2 *a
/ 2 + ----- + ------- + -------------------------
\/ a 2 ____
3/4 4 / 4 ___ 3/2
2 *\/ a - \/ 2 *a
$$\frac{- 2 \sqrt{a} + \sqrt[4]{2} a^{2}}{- \sqrt{2} a^{\frac{3}{2}} + 2^{\frac{3}{4}} \sqrt[4]{a^{4}}} + \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}}$$
sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) + (-2*sqrt(a) + 2^(1/4)*a^2)/(2^(3/4)*(a^4)^(1/4) - sqrt(2)*a^(3/2))
Potencias
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_____________________
/ ___ ___ ___ 4 ___ 2
/ \/ 2 a*\/ 2 - 2*\/ a + \/ 2 *a
/ 2 + ----- + ------- - -------------------------
\/ a 2 ____
___ 3/2 3/4 4 / 4
\/ 2 *a - 2 *\/ a
$$- \frac{- 2 \sqrt{a} + \sqrt[4]{2} a^{2}}{\sqrt{2} a^{\frac{3}{2}} - 2^{\frac{3}{4}} \sqrt[4]{a^{4}}} + \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}}$$
_____________________
/ ___ ___ ___ 4 ___ 2
/ \/ 2 a*\/ 2 2*\/ a - \/ 2 *a
/ 2 + ----- + ------- + -------------------------
\/ a 2 ____
___ 3/2 3/4 4 / 4
\/ 2 *a - 2 *\/ a
$$\frac{2 \sqrt{a} - \sqrt[4]{2} a^{2}}{\sqrt{2} a^{\frac{3}{2}} - 2^{\frac{3}{4}} \sqrt[4]{a^{4}}} + \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}}$$
sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) + (2*sqrt(a) - 2^(1/4)*a^2)/(sqrt(2)*a^(3/2) - 2^(3/4)*(a^4)^(1/4))
Unión de expresiones racionales
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______________________
/ ___ / 2\ / ____\
___ 4 ___ 2 ___ / 4*a + \/ 2 *\2 + a / | ___ 3/2 3/4 4 / 4 |
4*\/ a - 2*\/ 2 *a + \/ 2 * / -------------------- *\\/ 2 *a - 2 *\/ a /
\/ a
------------------------------------------------------------------------------------
/ ____\
| ___ 3/2 3/4 4 / 4 |
2*\\/ 2 *a - 2 *\/ a /
$$\frac{4 \sqrt{a} - 2 \sqrt[4]{2} a^{2} + \sqrt{2} \sqrt{\frac{4 a + \sqrt{2} \left(a^{2} + 2\right)}{a}} \left(\sqrt{2} a^{\frac{3}{2}} - 2^{\frac{3}{4}} \sqrt[4]{a^{4}}\right)}{2 \left(\sqrt{2} a^{\frac{3}{2}} - 2^{\frac{3}{4}} \sqrt[4]{a^{4}}\right)}$$
(4*sqrt(a) - 2*2^(1/4)*a^2 + sqrt(2)*sqrt((4*a + sqrt(2)*(2 + a^2))/a)*(sqrt(2)*a^(3/2) - 2^(3/4)*(a^4)^(1/4)))/(2*(sqrt(2)*a^(3/2) - 2^(3/4)*(a^4)^(1/4)))
Parte trigonométrica
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_____________________
/ ___ ___ ___ 4 ___ 2
/ \/ 2 a*\/ 2 - 2*\/ a + \/ 2 *a
/ 2 + ----- + ------- - -------------------------
\/ a 2 ____
___ 3/2 3/4 4 / 4
\/ 2 *a - 2 *\/ a
$$- \frac{- 2 \sqrt{a} + \sqrt[4]{2} a^{2}}{\sqrt{2} a^{\frac{3}{2}} - 2^{\frac{3}{4}} \sqrt[4]{a^{4}}} + \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}}$$
sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) - (-2*sqrt(a) + 2^(1/4)*a^2)/(sqrt(2)*a^(3/2) - 2^(3/4)*(a^4)^(1/4))
Combinatoria
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_______________________ _______________________
/ ___ ____ / ___
___ 4 ___ 2 3/2 / ___ 2*\/ 2 4 ___ 4 / 4 / ___ 2*\/ 2
- 2*\/ a + \/ 2 *a - a * / 4 + a*\/ 2 + ------- + \/ 2 *\/ a * / 4 + a*\/ 2 + -------
\/ a \/ a
-----------------------------------------------------------------------------------------------------
____
3/4 4 / 4 ___ 3/2
2 *\/ a - \/ 2 *a
$$\frac{- a^{\frac{3}{2}} \sqrt{\sqrt{2} a + 4 + \frac{2 \sqrt{2}}{a}} - 2 \sqrt{a} + \sqrt[4]{2} a^{2} + \sqrt[4]{2} \sqrt{\sqrt{2} a + 4 + \frac{2 \sqrt{2}}{a}} \sqrt[4]{a^{4}}}{- \sqrt{2} a^{\frac{3}{2}} + 2^{\frac{3}{4}} \sqrt[4]{a^{4}}}$$
(-2*sqrt(a) + 2^(1/4)*a^2 - a^(3/2)*sqrt(4 + a*sqrt(2) + 2*sqrt(2)/a) + 2^(1/4)*(a^4)^(1/4)*sqrt(4 + a*sqrt(2) + 2*sqrt(2)/a))/(2^(3/4)*(a^4)^(1/4) - sqrt(2)*a^(3/2))
Denominador racional
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_____________________ _____________________ _____________________ _____________________ _____________________ _____________________ _____________________ _____________________
____ ____ ____ ____ / ___ ___ ____ ____ ____ ____ ____ / ___ ___ ____ ____ / ___ ___ ____ / ___ ___ ____ / ___ ___ ____ / ___ ___ ____ ____ / ___ ___ ____ ____ / ___ ___
5 4 / 4 5/2 / 3 3/4 5 / 3 2 / 4 / \/ 2 a*\/ 2 ___ 4 4 / 4 4 ___ 4 / 3 ___ 7/2 / 3 3/4 7/2 4 / 4 9/2 / 3 / \/ 2 a*\/ 2 4 ___ 5/2 4 / 4 ___ 3 / 4 / \/ 2 a*\/ 2 ___ 7/2 / 3 / \/ 2 a*\/ 2 4 ___ 9/2 4 / 4 / \/ 2 a*\/ 2 3/4 7/2 4 / 4 / \/ 2 a*\/ 2 4 ___ 3 / 3 4 / 4 / \/ 2 a*\/ 2 3/4 2 / 3 4 / 4 / \/ 2 a*\/ 2
- 2*a *\/ a + 4*a *\/ a - 2 *a *\/ a - 4*a *\/ a * / 2 + ----- + ------- - 2*\/ 2 *a *\/ a - 2*\/ 2 *a *\/ a + 2*\/ 2 *a *\/ a + 2*2 *a *\/ a + 2*a *\/ a * / 2 + ----- + ------- + 4*\/ 2 *a *\/ a - 2*\/ 2 *a *\/ a * / 2 + ----- + ------- + 2*\/ 2 *a *\/ a * / 2 + ----- + ------- + 2*\/ 2 *a *\/ a * / 2 + ----- + ------- + 2*2 *a *\/ a * / 2 + ----- + ------- - 2*\/ 2 *a *\/ a *\/ a * / 2 + ----- + ------- - 2*2 *a *\/ a *\/ a * / 2 + ----- + -------
\/ a 2 \/ a 2 \/ a 2 \/ a 2 \/ a 2 \/ a 2 \/ a 2 \/ a 2
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
4 / 2\
2*a *\-2 + a /
$$\frac{2 a^{\frac{9}{2}} \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}} \sqrt{a^{3}} + 2 \sqrt[4]{2} a^{\frac{9}{2}} \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}} \sqrt[4]{a^{4}} + 2 \sqrt{2} a^{\frac{7}{2}} \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}} \sqrt{a^{3}} + 2 \cdot 2^{\frac{3}{4}} a^{\frac{7}{2}} \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}} \sqrt[4]{a^{4}} + 2 \sqrt{2} a^{\frac{7}{2}} \sqrt{a^{3}} + 2 \cdot 2^{\frac{3}{4}} a^{\frac{7}{2}} \sqrt[4]{a^{4}} + 4 a^{\frac{5}{2}} \sqrt{a^{3}} + 4 \sqrt[4]{2} a^{\frac{5}{2}} \sqrt[4]{a^{4}} - 2^{\frac{3}{4}} a^{5} \sqrt{a^{3}} - 2 a^{5} \sqrt[4]{a^{4}} - 2 \sqrt[4]{2} a^{4} \sqrt{a^{3}} - 2 \sqrt{2} a^{4} \sqrt[4]{a^{4}} - 2 \sqrt[4]{2} a^{3} \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}} \sqrt{a^{3}} \sqrt[4]{a^{4}} - 2 \sqrt{2} a^{3} \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}} \sqrt{a^{4}} - 2 \cdot 2^{\frac{3}{4}} a^{2} \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}} \sqrt{a^{3}} \sqrt[4]{a^{4}} - 4 a^{2} \sqrt{\frac{\sqrt{2} a}{2} + 2 + \frac{\sqrt{2}}{a}} \sqrt{a^{4}}}{2 a^{4} \left(a^{2} - 2\right)}$$
(-2*a^5*(a^4)^(1/4) + 4*a^(5/2)*sqrt(a^3) - 2^(3/4)*a^5*sqrt(a^3) - 4*a^2*sqrt(a^4)*sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) - 2*sqrt(2)*a^4*(a^4)^(1/4) - 2*2^(1/4)*a^4*sqrt(a^3) + 2*sqrt(2)*a^(7/2)*sqrt(a^3) + 2*2^(3/4)*a^(7/2)*(a^4)^(1/4) + 2*a^(9/2)*sqrt(a^3)*sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) + 4*2^(1/4)*a^(5/2)*(a^4)^(1/4) - 2*sqrt(2)*a^3*sqrt(a^4)*sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) + 2*sqrt(2)*a^(7/2)*sqrt(a^3)*sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) + 2*2^(1/4)*a^(9/2)*(a^4)^(1/4)*sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) + 2*2^(3/4)*a^(7/2)*(a^4)^(1/4)*sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) - 2*2^(1/4)*a^3*sqrt(a^3)*(a^4)^(1/4)*sqrt(2 + sqrt(2)/a + a*sqrt(2)/2) - 2*2^(3/4)*a^2*sqrt(a^3)*(a^4)^(1/4)*sqrt(2 + sqrt(2)/a + a*sqrt(2)/2))/(2*a^4*(-2 + a^2))