Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sqrt((sqrt(a)+1/(sqrt(a)+2))*(sqrt(a)+1/(sqrt(a)-2))*(a-4))+a
¿Cómo vas a descomponer esta sqrt((sqrt(a)+1/(sqrt(a)+2))*(sqrt(a)+1/(sqrt(a)-2))*(a-4))+a expresión en fracciones?
Solución
Ha introducido
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/ / ___ 1 \ / ___ 1 \
/ |\/ a + ---------|*|\/ a + ---------|*(a - 4) + a
/ | ___ | | ___ |
\/ \ \/ a + 2/ \ \/ a - 2/
$$a + \sqrt{\left(\sqrt{a} + \frac{1}{\sqrt{a} - 2}\right) \left(\sqrt{a} + \frac{1}{\sqrt{a} + 2}\right) \left(a - 4\right)}$$
sqrt(((sqrt(a) + 1/(sqrt(a) + 2))*(sqrt(a) + 1/(sqrt(a) - 2)))*(a - 4)) + a
Simplificación general
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______________________
/ 3 2
/ -4 + a - 6*a + 9*a
a + / --------------------
\/ -4 + a
$$a + \sqrt{\frac{a^{3} - 6 a^{2} + 9 a - 4}{a - 4}}$$
a + sqrt((-4 + a^3 - 6*a^2 + 9*a)/(-4 + a))
Respuesta numérica
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a + ((-4.0 + a)*(a^0.5 + 1/(2.0 + a^0.5))*(a^0.5 + 1/(-2.0 + a^0.5)))^0.5
a + ((-4.0 + a)*(a^0.5 + 1/(2.0 + a^0.5))*(a^0.5 + 1/(-2.0 + a^0.5)))^0.5
Denominador racional
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/ 2 2
/ 2 16 a 8*a 8*a 2*a
a + / a - 4*a + --------- + --------- - ------ - --------- + ------
/ 2 2 -4 + a 2 -4 + a
\/ (-4 + a) (-4 + a) (-4 + a)
$$a + \sqrt{a^{2} + \frac{2 a^{2}}{a - 4} + \frac{a^{2}}{\left(a - 4\right)^{2}} - 4 a - \frac{8 a}{a - 4} - \frac{8 a}{\left(a - 4\right)^{2}} + \frac{16}{\left(a - 4\right)^{2}}}$$
a + sqrt(a^2 - 4*a + 16/(-4 + a)^2 + a^2/(-4 + a)^2 - 8*a/(-4 + a) - 8*a/(-4 + a)^2 + 2*a^2/(-4 + a))
Unión de expresiones racionales
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/ / ___ / ___\\ / ___ / ___\\
/ \1 + \/ a *\-2 + \/ a //*\1 + \/ a *\2 + \/ a //*(-4 + a)
a + / ---------------------------------------------------------
/ / ___\ / ___\
\/ \-2 + \/ a /*\2 + \/ a /
$$a + \sqrt{\frac{\left(a - 4\right) \left(\sqrt{a} \left(\sqrt{a} - 2\right) + 1\right) \left(\sqrt{a} \left(\sqrt{a} + 2\right) + 1\right)}{\left(\sqrt{a} - 2\right) \left(\sqrt{a} + 2\right)}}$$
a + sqrt((1 + sqrt(a)*(-2 + sqrt(a)))*(1 + sqrt(a)*(2 + sqrt(a)))*(-4 + a)/((-2 + sqrt(a))*(2 + sqrt(a))))
Denominador común
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/ 3/2 3/2 ___ ___
/ 2 4 a a a 4*\/ a 4*\/ a
a + / a - 4*a - ------ + ------ + ---------- + --------- - ---------- - ---------
/ -4 + a -4 + a ___ ___ ___ ___
\/ -2 + \/ a 2 + \/ a -2 + \/ a 2 + \/ a
$$a + \sqrt{\frac{a^{\frac{3}{2}}}{\sqrt{a} + 2} + \frac{a^{\frac{3}{2}}}{\sqrt{a} - 2} - \frac{4 \sqrt{a}}{\sqrt{a} + 2} - \frac{4 \sqrt{a}}{\sqrt{a} - 2} + a^{2} - 4 a + \frac{a}{a - 4} - \frac{4}{a - 4}}$$
a + sqrt(a^2 - 4*a - 4/(-4 + a) + a/(-4 + a) + a^(3/2)/(-2 + sqrt(a)) + a^(3/2)/(2 + sqrt(a)) - 4*sqrt(a)/(-2 + sqrt(a)) - 4*sqrt(a)/(2 + sqrt(a)))
Combinatoria
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/ 3 2
/ 4 a 6*a 9*a
a + / - ------ + ------ - ------ + ------
\/ -4 + a -4 + a -4 + a -4 + a
$$a + \sqrt{\frac{a^{3}}{a - 4} - \frac{6 a^{2}}{a - 4} + \frac{9 a}{a - 4} - \frac{4}{a - 4}}$$
a + sqrt(-4/(-4 + a) + a^3/(-4 + a) - 6*a^2/(-4 + a) + 9*a/(-4 + a))