展開する (0.3+0.7e^t)^5

解

展開する

(0.3 + 0.7 e^{t})^{5}

0.00243 + 0.02835 e^{t} + 0.1323 e^{2 t} + 0.3087 e^{3 t} + 0.36015 e^{4 t} + 0.16807 e^{5 t}

解答ステップ

(0.3 + 0.7 e^{t})^{5}

2項定理を適用する:

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

a = 0.3, b = 0.7 e^{t}

= i = 0 \sum 5 (i 5) \cdot 0. 3^{(5 - i)} (0.7 e^{t})^{i}

総和を展開する

= \frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 0. 3^{5} (0.7 e^{t})^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 0. 3^{4} (0.7 e^{t})^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 0. 3^{3} (0.7 e^{t})^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 0. 3^{2} (0.7 e^{t})^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 0. 3^{1} (0.7 e^{t})^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 0. 3^{0} (0.7 e^{t})^{5}

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 0. 3^{5} (0.7 e^{t})^{0} = 0.00243

簡素化

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 0. 3^{4} (0.7 e^{t})^{1} : 0.02835 e^{t}

簡素化

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 0. 3^{3} (0.7 e^{t})^{2} : 0.1323 e^{2 t}

簡素化

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 0. 3^{2} (0.7 e^{t})^{3} : 0.3087 e^{3 t}

簡素化

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 0. 3^{1} (0.7 e^{t})^{4} : 0.36015 e^{4 t}

簡素化

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 0. 3^{0} (0.7 e^{t})^{5} : 0.16807 e^{5 t}

= 0.00243 + 0.02835 e^{t} + 0.1323 e^{2 t} + 0.3087 e^{3 t} + 0.36015 e^{4 t} + 0.16807 e^{5 t}