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Popular Algebra >

expand (e^x+1)^{15}

Solution

expand

(e^{x} + 1)^{15}

Solution

e^{15 x} + 15 e^{14 x} + 105 e^{13 x} + 455 e^{12 x} + 1365 e^{11 x} + 3003 e^{10 x} + 5005 e^{9 x} + 6435 e^{8 x} + 6435 e^{7 x} + 5005 e^{6 x} + 3003 e^{5 x} + 1365 e^{4 x} + 455 e^{3 x} + 105 e^{2 x} + 15 e^{x} + 1

Solution steps

(e^{x} + 1)^{15}

Apply binomial theorem:

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

a = e^{x}, b = 1

= i = 0 \sum 15 (i 15) (e^{x})^{(15 - i)} \cdot 1^{i}

Expand summation

= \frac{15 !}{0 ! ( 15 - 0 ) !} (e^{x})^{15} \cdot 1^{0} + \frac{15 !}{1 ! ( 15 - 1 ) !} (e^{x})^{14} \cdot 1^{1} + \frac{15 !}{2 ! ( 15 - 2 ) !} (e^{x})^{13} \cdot 1^{2} + \frac{15 !}{3 ! ( 15 - 3 ) !} (e^{x})^{12} \cdot 1^{3} + \frac{15 !}{4 ! ( 15 - 4 ) !} (e^{x})^{11} \cdot 1^{4} + \frac{15 !}{5 ! ( 15 - 5 ) !} (e^{x})^{10} \cdot 1^{5} + \frac{15 !}{6 ! ( 15 - 6 ) !} (e^{x})^{9} \cdot 1^{6} + \frac{15 !}{7 ! ( 15 - 7 ) !} (e^{x})^{8} \cdot 1^{7} + \frac{15 !}{8 ! ( 15 - 8 ) !} (e^{x})^{7} \cdot 1^{8} + \frac{15 !}{9 ! ( 15 - 9 ) !} (e^{x})^{6} \cdot 1^{9} + \frac{15 !}{10 ! ( 15 - 10 ) !} (e^{x})^{5} \cdot 1^{10} + \frac{15 !}{11 ! ( 15 - 11 ) !} (e^{x})^{4} \cdot 1^{11} + \frac{15 !}{12 ! ( 15 - 12 ) !} (e^{x})^{3} \cdot 1^{12} + \frac{15 !}{13 ! ( 15 - 13 ) !} (e^{x})^{2} \cdot 1^{13} + \frac{15 !}{14 ! ( 15 - 14 ) !} (e^{x})^{1} \cdot 1^{14} + \frac{15 !}{15 ! ( 15 - 15 ) !} (e^{x})^{0} \cdot 1^{15}

Simplify

\frac{15 !}{0 ! ( 15 - 0 ) !} (e^{x})^{15} \cdot 1^{0} : e^{15 x}

Simplify

\frac{15 !}{1 ! ( 15 - 1 ) !} (e^{x})^{14} \cdot 1^{1} : 15 e^{14 x}

Simplify

\frac{15 !}{2 ! ( 15 - 2 ) !} (e^{x})^{13} \cdot 1^{2} : 105 e^{13 x}

Simplify

\frac{15 !}{3 ! ( 15 - 3 ) !} (e^{x})^{12} \cdot 1^{3} : 455 e^{12 x}

Simplify

\frac{15 !}{4 ! ( 15 - 4 ) !} (e^{x})^{11} \cdot 1^{4} : 1365 e^{11 x}

Simplify

\frac{15 !}{5 ! ( 15 - 5 ) !} (e^{x})^{10} \cdot 1^{5} : 3003 e^{10 x}

Simplify

\frac{15 !}{6 ! ( 15 - 6 ) !} (e^{x})^{9} \cdot 1^{6} : 5005 e^{9 x}

Simplify

\frac{15 !}{7 ! ( 15 - 7 ) !} (e^{x})^{8} \cdot 1^{7} : 6435 e^{8 x}

Simplify

\frac{15 !}{8 ! ( 15 - 8 ) !} (e^{x})^{7} \cdot 1^{8} : 6435 e^{7 x}

Simplify

\frac{15 !}{9 ! ( 15 - 9 ) !} (e^{x})^{6} \cdot 1^{9} : 5005 e^{6 x}

Simplify

\frac{15 !}{10 ! ( 15 - 10 ) !} (e^{x})^{5} \cdot 1^{10} : 3003 e^{5 x}

Simplify

\frac{15 !}{11 ! ( 15 - 11 ) !} (e^{x})^{4} \cdot 1^{11} : 1365 e^{4 x}

Simplify

\frac{15 !}{12 ! ( 15 - 12 ) !} (e^{x})^{3} \cdot 1^{12} : 455 e^{3 x}

Simplify

\frac{15 !}{13 ! ( 15 - 13 ) !} (e^{x})^{2} \cdot 1^{13} : 105 e^{2 x}

Simplify

\frac{15 !}{14 ! ( 15 - 14 ) !} (e^{x})^{1} \cdot 1^{14} : 15 e^{x}

\frac{15 !}{15 ! ( 15 - 15 ) !} (e^{x})^{0} \cdot 1^{15} = 1

= e^{15 x} + 15 e^{14 x} + 105 e^{13 x} + 455 e^{12 x} + 1365 e^{11 x} + 3003 e^{10 x} + 5005 e^{9 x} + 6435 e^{8 x} + 6435 e^{7 x} + 5005 e^{6 x} + 3003 e^{5 x} + 1365 e^{4 x} + 455 e^{3 x} + 105 e^{2 x} + 15 e^{x} + 1

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