Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(a*sqrt(a)+b*sqrt(b))/(sqrt(a)+sqrt(b))/(a-b)+(2*sqrt(b)/sqrt(a)+sqrt(b))
¿Cómo vas a descomponer esta (a*sqrt(a)+b*sqrt(b))/(sqrt(a)+sqrt(b))/(a-b)+(2*sqrt(b)/sqrt(a)+sqrt(b)) expresión en fracciones?
Solución
Ha introducido
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/ ___ ___\
|a*\/ a + b*\/ b |
|-----------------|
| ___ ___ | ___
\ \/ a + \/ b / 2*\/ b ___
------------------- + ------- + \/ b
a - b ___
\/ a
$$\frac{\frac{1}{\sqrt{a} + \sqrt{b}} \left(\sqrt{a} a + \sqrt{b} b\right)}{a - b} + \left(\sqrt{b} + \frac{2 \sqrt{b}}{\sqrt{a}}\right)$$
((a*sqrt(a) + b*sqrt(b))/(sqrt(a) + sqrt(b)))/(a - b) + (2*sqrt(b))/sqrt(a) + sqrt(b)
Simplificación general
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___ 3/2 3/2
___ 2*\/ b a + b
\/ b + ------- + -----------------------
___ / ___ ___\
\/ a (a - b)*\\/ a + \/ b /
$$\sqrt{b} + \frac{a^{\frac{3}{2}} + b^{\frac{3}{2}}}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)} + \frac{2 \sqrt{b}}{\sqrt{a}}$$
sqrt(b) + 2*sqrt(b)/sqrt(a) + (a^(3/2) + b^(3/2))/((a - b)*(sqrt(a) + sqrt(b)))
Respuesta numérica
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b^0.5 + 2.0*a^(-0.5)*b^0.5 + (a^1.5 + b^1.5)/((a - b)*(a^0.5 + b^0.5))
b^0.5 + 2.0*a^(-0.5)*b^0.5 + (a^1.5 + b^1.5)/((a - b)*(a^0.5 + b^0.5))
Compilar la expresión
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___ 3/2 3/2
___ 2*\/ b a + b
\/ b + ------- + -----------------------
___ / ___ ___\
\/ a (a - b)*\\/ a + \/ b /
$$\sqrt{b} + \frac{a^{\frac{3}{2}} + b^{\frac{3}{2}}}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)} + \frac{2 \sqrt{b}}{\sqrt{a}}$$
3/2 3/2
___ / 2 \ a + b
\/ b *|1 + -----| + -----------------------
| ___| / ___ ___\
\ \/ a / (a - b)*\\/ a + \/ b /
$$\sqrt{b} \left(1 + \frac{2}{\sqrt{a}}\right) + \frac{a^{\frac{3}{2}} + b^{\frac{3}{2}}}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}$$
sqrt(b)*(1 + 2/sqrt(a)) + (a^(3/2) + b^(3/2))/((a - b)*(sqrt(a) + sqrt(b)))
Denominador racional
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5/2 5/2 ___ 5/2 2 ___ 5/2 ___ ___ 2 3/2 3/2 3/2
a + 2*b + \/ a *b + a *\/ b + a *\/ b - \/ a *b - 3*a*b - 2*a *b
--------------------------------------------------------------------------------------
___ 2
\/ a *(a - b)
$$\frac{a^{\frac{5}{2}} \sqrt{b} + a^{\frac{5}{2}} - 2 a^{\frac{3}{2}} b^{\frac{3}{2}} + \sqrt{a} b^{\frac{5}{2}} - \sqrt{a} b^{2} + a^{2} \sqrt{b} - 3 a b^{\frac{3}{2}} + 2 b^{\frac{5}{2}}}{\sqrt{a} \left(a - b\right)^{2}}$$
(a^(5/2) + 2*b^(5/2) + sqrt(a)*b^(5/2) + a^2*sqrt(b) + a^(5/2)*sqrt(b) - sqrt(a)*b^2 - 3*a*b^(3/2) - 2*a^(3/2)*b^(3/2))/(sqrt(a)*(a - b)^2)
Unión de expresiones racionales
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___ / 3/2 3/2\ ___ / ___\ / ___ ___\
\/ a *\a + b / + \/ b *\2 + \/ a /*(a - b)*\\/ a + \/ b /
---------------------------------------------------------------
___ / ___ ___\
\/ a *(a - b)*\\/ a + \/ b /
$$\frac{\sqrt{a} \left(a^{\frac{3}{2}} + b^{\frac{3}{2}}\right) + \sqrt{b} \left(\sqrt{a} + 2\right) \left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}{\sqrt{a} \left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}$$
(sqrt(a)*(a^(3/2) + b^(3/2)) + sqrt(b)*(2 + sqrt(a))*(a - b)*(sqrt(a) + sqrt(b)))/(sqrt(a)*(a - b)*(sqrt(a) + sqrt(b)))
Combinatoria
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2 2 3/2 2 ___ 3/2 ___ 2 ___ 3/2 3/2 ___
a - 2*b + b*a + a *\/ b - a*b - \/ a *b - \/ a *b + 2*a*b + 2*a *\/ b
-------------------------------------------------------------------------------------
___ / ___ ___\
\/ a *(a - b)*\\/ a + \/ b /
$$\frac{2 a^{\frac{3}{2}} \sqrt{b} + a^{\frac{3}{2}} b - \sqrt{a} b^{\frac{3}{2}} - \sqrt{a} b^{2} + a^{2} \sqrt{b} + a^{2} - a b^{\frac{3}{2}} + 2 a b - 2 b^{2}}{\sqrt{a} \left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}$$
(a^2 - 2*b^2 + b*a^(3/2) + a^2*sqrt(b) - a*b^(3/2) - sqrt(a)*b^2 - sqrt(a)*b^(3/2) + 2*a*b + 2*a^(3/2)*sqrt(b))/(sqrt(a)*(a - b)*(sqrt(a) + sqrt(b)))
Potencias
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___ 3/2 3/2
___ 2*\/ b a + b
\/ b + ------- + -----------------------
___ / ___ ___\
\/ a (a - b)*\\/ a + \/ b /
$$\sqrt{b} + \frac{a^{\frac{3}{2}} + b^{\frac{3}{2}}}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)} + \frac{2 \sqrt{b}}{\sqrt{a}}$$
sqrt(b) + 2*sqrt(b)/sqrt(a) + (a^(3/2) + b^(3/2))/((a - b)*(sqrt(a) + sqrt(b)))
Parte trigonométrica
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___ 3/2 3/2
___ 2*\/ b a + b
\/ b + ------- + -----------------------
___ / ___ ___\
\/ a (a - b)*\\/ a + \/ b /
$$\sqrt{b} + \frac{a^{\frac{3}{2}} + b^{\frac{3}{2}}}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)} + \frac{2 \sqrt{b}}{\sqrt{a}}$$
sqrt(b) + 2*sqrt(b)/sqrt(a) + (a^(3/2) + b^(3/2))/((a - b)*(sqrt(a) + sqrt(b)))
Denominador común
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2 3/2 ___
___ - 2*b + a *\/ b + 3*a*b
1 + \/ b + ----------------------------------
2 3/2 ___ ___ 3/2
a + a *\/ b - a*b - \/ a *b
$$\sqrt{b} + \frac{a^{\frac{3}{2}} \sqrt{b} + 3 a b - 2 b^{2}}{a^{\frac{3}{2}} \sqrt{b} - \sqrt{a} b^{\frac{3}{2}} + a^{2} - a b} + 1$$
1 + sqrt(b) + (-2*b^2 + a^(3/2)*sqrt(b) + 3*a*b)/(a^2 + a^(3/2)*sqrt(b) - a*b - sqrt(a)*b^(3/2))