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Popular Algebra >

expand (a+b)^{15}

Solution

expand

(a + b)^{15}

Solution

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

Solution steps

(a + b)^{15}

Apply binomial theorem:

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

a = a, b = b

= i = 0 \sum 15 (i 15) a^{(15 - i)} b^{i}

Expand summation

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

Simplify

\frac{15 !}{0 ! ( 15 - 0 ) !} a^{15} b^{0} : a^{15}

Simplify

\frac{15 !}{1 ! ( 15 - 1 ) !} a^{14} b^{1} : 15 a^{14} b

Simplify

\frac{15 !}{2 ! ( 15 - 2 ) !} a^{13} b^{2} : 105 a^{13} b^{2}

Simplify

\frac{15 !}{3 ! ( 15 - 3 ) !} a^{12} b^{3} : 455 a^{12} b^{3}

Simplify

\frac{15 !}{4 ! ( 15 - 4 ) !} a^{11} b^{4} : 1365 a^{11} b^{4}

Simplify

\frac{15 !}{5 ! ( 15 - 5 ) !} a^{10} b^{5} : 3003 a^{10} b^{5}

Simplify

\frac{15 !}{6 ! ( 15 - 6 ) !} a^{9} b^{6} : 5005 a^{9} b^{6}

Simplify

\frac{15 !}{7 ! ( 15 - 7 ) !} a^{8} b^{7} : 6435 a^{8} b^{7}

Simplify

\frac{15 !}{8 ! ( 15 - 8 ) !} a^{7} b^{8} : 6435 a^{7} b^{8}

Simplify

\frac{15 !}{9 ! ( 15 - 9 ) !} a^{6} b^{9} : 5005 a^{6} b^{9}

Simplify

\frac{15 !}{10 ! ( 15 - 10 ) !} a^{5} b^{10} : 3003 a^{5} b^{10}

Simplify

\frac{15 !}{11 ! ( 15 - 11 ) !} a^{4} b^{11} : 1365 a^{4} b^{11}

Simplify

\frac{15 !}{12 ! ( 15 - 12 ) !} a^{3} b^{12} : 455 a^{3} b^{12}

Simplify

\frac{15 !}{13 ! ( 15 - 13 ) !} a^{2} b^{13} : 105 a^{2} b^{13}

Simplify

\frac{15 !}{14 ! ( 15 - 14 ) !} a^{1} b^{14} : 15 a b^{14}

Simplify

\frac{15 !}{15 ! ( 15 - 15 ) !} a^{0} b^{15} : b^{15}

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

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