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desarrollar (1/4+3/4 e^t)^5

Solución

desarrollar

(\frac{1}{4} + \frac{3}{4} e^{t})^{5}

Solución

\frac{1}{1024} + \frac{15 e ^{t}}{1024} + \frac{243 e ^{5 t}}{1024} + \frac{45 e ^{2 t}}{512} + \frac{135 e ^{3 t}}{512} + \frac{405}{1024} e^{4 t}

Pasos de solución

(\frac{1}{4} + \frac{3}{4} e^{t})^{5}

Aplicar el teorema del binomio:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = \frac{1}{4}, b = \frac{3}{4} e^{t}

= i = 0 \sum 5 (i 5) (\frac{1}{4})^{(5 - i)} (\frac{3}{4} e^{t})^{i}

Expandir sumatorio

= \frac{5 !}{0 ! ( 5 - 0 ) !} (\frac{1}{4})^{5} (\frac{3}{4} e^{t})^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} (\frac{1}{4})^{4} (\frac{3}{4} e^{t})^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} (\frac{1}{4})^{3} (\frac{3}{4} e^{t})^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} (\frac{1}{4})^{2} (\frac{3}{4} e^{t})^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} (\frac{1}{4})^{1} (\frac{3}{4} e^{t})^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} (\frac{1}{4})^{0} (\frac{3}{4} e^{t})^{5}

\frac{5 !}{0 ! ( 5 - 0 ) !} (\frac{1}{4})^{5} (\frac{3}{4} e^{t})^{0} = \frac{1}{1024}

Simplificar

\frac{5 !}{1 ! ( 5 - 1 ) !} (\frac{1}{4})^{4} (\frac{3}{4} e^{t})^{1} : \frac{15 e ^{t}}{1024}

Simplificar

\frac{5 !}{2 ! ( 5 - 2 ) !} (\frac{1}{4})^{3} (\frac{3}{4} e^{t})^{2} : \frac{45 e ^{2 t}}{512}

Simplificar

\frac{5 !}{3 ! ( 5 - 3 ) !} (\frac{1}{4})^{2} (\frac{3}{4} e^{t})^{3} : \frac{135 e ^{3 t}}{512}

Simplificar

\frac{5 !}{4 ! ( 5 - 4 ) !} (\frac{1}{4})^{1} (\frac{3}{4} e^{t})^{4} : \frac{405}{1024} e^{4 t}

Simplificar

\frac{5 !}{5 ! ( 5 - 5 ) !} (\frac{1}{4})^{0} (\frac{3}{4} e^{t})^{5} : \frac{243 e ^{5 t}}{1024}

= \frac{1}{1024} + \frac{15 e ^{t}}{1024} + \frac{45 e ^{2 t}}{512} + \frac{135 e ^{3 t}}{512} + \frac{405}{1024} e^{4 t} + \frac{243 e ^{5 t}}{1024}

Simplificar

\frac{1}{1024} + \frac{15 e ^{t}}{1024} + \frac{45 e ^{2 t}}{512} + \frac{135 e ^{3 t}}{512} + \frac{405}{1024} e^{4 t} + \frac{243 e ^{5 t}}{1024} : \frac{1 + 15 e ^{t} + 243 e ^{5 t}}{1024} + \frac{45 e ^{2 t} + 135 e ^{3 t}}{512} + \frac{405}{1024} e^{4 t}

= \frac{1 + 15 e ^{t} + 243 e ^{5 t}}{1024} + \frac{45 e ^{2 t} + 135 e ^{3 t}}{512} + \frac{405}{1024} e^{4 t}

Aplicar las propiedades de las fracciones:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 + 15 e ^{t} + 243 e ^{5 t}}{1024} = \frac{1}{1024} + \frac{15 e ^{t}}{1024} + \frac{243 e ^{5 t}}{1024}

= \frac{1}{1024} + \frac{15 e ^{t}}{1024} + \frac{243 e ^{5 t}}{1024} + \frac{45 e ^{2 t} + 135 e ^{3 t}}{512} + \frac{405}{1024} e^{4 t}

Aplicar las propiedades de las fracciones:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{45 e ^{2 t} + 135 e ^{3 t}}{512} = \frac{45 e ^{2 t}}{512} + \frac{135 e ^{3 t}}{512}

= \frac{1}{1024} + \frac{15 e ^{t}}{1024} + \frac{243 e ^{5 t}}{1024} + \frac{45 e ^{2 t}}{512} + \frac{135 e ^{3 t}}{512} + \frac{405}{1024} e^{4 t}

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