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
------------------- + -------------
a - b ___ ___
\/ a + \/ b
$$\frac{2 \sqrt{b}}{\sqrt{a} + \sqrt{b}} + \frac{\frac{1}{\sqrt{a} + \sqrt{b}} \left(\sqrt{a} a + \sqrt{b} b\right)}{a - b}$$
((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 ___
a + b + 2*\/ b *(a - b)
-----------------------------
/ ___ ___\
(a - b)*\\/ a + \/ b /
$$\frac{a^{\frac{3}{2}} + b^{\frac{3}{2}} + 2 \sqrt{b} \left(a - b\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}$$
(a^(3/2) + b^(3/2) + 2*sqrt(b)*(a - b))/((a - b)*(sqrt(a) + sqrt(b)))
Denominador racional
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2 2 2 2 2
2 / ___ ___\ 2 / ___ ___\ ___ 3/2 / ___ ___\ / ___ ___\ 3/2 ___ / ___ ___\
a *\\/ a - \/ b / - b *\\/ a - \/ b / - \/ a *b *\\/ a - \/ b / + 2*a*b*\\/ a - \/ b / + 3*a *\/ b *\\/ a - \/ b /
--------------------------------------------------------------------------------------------------------------------------------
3
(a - b)
$$\frac{3 a^{\frac{3}{2}} \sqrt{b} \left(\sqrt{a} - \sqrt{b}\right)^{2} - \sqrt{a} b^{\frac{3}{2}} \left(\sqrt{a} - \sqrt{b}\right)^{2} + a^{2} \left(\sqrt{a} - \sqrt{b}\right)^{2} + 2 a b \left(\sqrt{a} - \sqrt{b}\right)^{2} - b^{2} \left(\sqrt{a} - \sqrt{b}\right)^{2}}{\left(a - b\right)^{3}}$$
(a^2*(sqrt(a) - sqrt(b))^2 - b^2*(sqrt(a) - sqrt(b))^2 - sqrt(a)*b^(3/2)*(sqrt(a) - sqrt(b))^2 + 2*a*b*(sqrt(a) - sqrt(b))^2 + 3*a^(3/2)*sqrt(b)*(sqrt(a) - sqrt(b))^2)/(a - b)^3
Respuesta numérica
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2.0*b^0.5/(a^0.5 + b^0.5) + (a^1.5 + b^1.5)/((a - b)*(a^0.5 + b^0.5))
2.0*b^0.5/(a^0.5 + b^0.5) + (a^1.5 + b^1.5)/((a - b)*(a^0.5 + b^0.5))
Denominador común
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3/2 3/2 ___
b - a + 2*b*\/ a
2 + -------------------------------
3/2 3/2 ___ ___
a - b + a*\/ b - b*\/ a
$$\frac{- a^{\frac{3}{2}} + 2 \sqrt{a} b + b^{\frac{3}{2}}}{a^{\frac{3}{2}} - \sqrt{a} b + a \sqrt{b} - b^{\frac{3}{2}}} + 2$$
2 + (b^(3/2) - a^(3/2) + 2*b*sqrt(a))/(a^(3/2) - b^(3/2) + a*sqrt(b) - b*sqrt(a))
Parte trigonométrica
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___ 3/2 3/2 2*\/ b a + b ------------- + ----------------------- ___ ___ / ___ ___\ \/ a + \/ b (a - b)*\\/ a + \/ b /
$$\frac{2 \sqrt{b}}{\sqrt{a} + \sqrt{b}} + \frac{a^{\frac{3}{2}} + b^{\frac{3}{2}}}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}$$
2*sqrt(b)/(sqrt(a) + sqrt(b)) + (a^(3/2) + b^(3/2))/((a - b)*(sqrt(a) + sqrt(b)))
Compilar la expresión
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___ 3/2 3/2 2*\/ b a + b ------------- + ----------------------- ___ ___ / ___ ___\ \/ a + \/ b (a - b)*\\/ a + \/ b /
$$\frac{2 \sqrt{b}}{\sqrt{a} + \sqrt{b}} + \frac{a^{\frac{3}{2}} + b^{\frac{3}{2}}}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}$$
2*sqrt(b)/(sqrt(a) + sqrt(b)) + (a^(3/2) + b^(3/2))/((a - b)*(sqrt(a) + sqrt(b)))
Unión de expresiones racionales
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3/2 3/2 ___
a + b + 2*\/ b *(a - b)
-----------------------------
/ ___ ___\
(a - b)*\\/ a + \/ b /
$$\frac{a^{\frac{3}{2}} + b^{\frac{3}{2}} + 2 \sqrt{b} \left(a - b\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}$$
(a^(3/2) + b^(3/2) + 2*sqrt(b)*(a - b))/((a - b)*(sqrt(a) + sqrt(b)))
Combinatoria
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3/2 3/2 ___
a - b + 2*a*\/ b
-----------------------
/ ___ ___\
(a - b)*\\/ a + \/ b /
$$\frac{a^{\frac{3}{2}} + 2 a \sqrt{b} - b^{\frac{3}{2}}}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}$$
(a^(3/2) - b^(3/2) + 2*a*sqrt(b))/((a - b)*(sqrt(a) + sqrt(b)))
Potencias
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___ 3/2 3/2 2*\/ b a + b ------------- + ----------------------- ___ ___ / ___ ___\ \/ a + \/ b (a - b)*\\/ a + \/ b /
$$\frac{2 \sqrt{b}}{\sqrt{a} + \sqrt{b}} + \frac{a^{\frac{3}{2}} + b^{\frac{3}{2}}}{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right)}$$
2*sqrt(b)/(sqrt(a) + sqrt(b)) + (a^(3/2) + b^(3/2))/((a - b)*(sqrt(a) + sqrt(b)))