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Popular Algebra >

expand (a+b)^{14}

Solution

expand

(a + b)^{14}

Solution

a^{14} + 14 a^{13} b + 91 a^{12} b^{2} + 364 a^{11} b^{3} + 1001 a^{10} b^{4} + 2002 a^{9} b^{5} + 3003 a^{8} b^{6} + 3432 a^{7} b^{7} + 3003 a^{6} b^{8} + 2002 a^{5} b^{9} + 1001 a^{4} b^{10} + 364 a^{3} b^{11} + 91 a^{2} b^{12} + 14 a b^{13} + b^{14}

Solution steps

(a + b)^{14}

Apply binomial theorem:

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

a = a, b = b

= i = 0 \sum 14 (i 14) a^{(14 - i)} b^{i}

Expand summation

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

Simplify

\frac{14 !}{0 ! ( 14 - 0 ) !} a^{14} b^{0} : a^{14}

Simplify

\frac{14 !}{1 ! ( 14 - 1 ) !} a^{13} b^{1} : 14 a^{13} b

Simplify

\frac{14 !}{2 ! ( 14 - 2 ) !} a^{12} b^{2} : 91 a^{12} b^{2}

Simplify

\frac{14 !}{3 ! ( 14 - 3 ) !} a^{11} b^{3} : 364 a^{11} b^{3}

Simplify

\frac{14 !}{4 ! ( 14 - 4 ) !} a^{10} b^{4} : 1001 a^{10} b^{4}

Simplify

\frac{14 !}{5 ! ( 14 - 5 ) !} a^{9} b^{5} : 2002 a^{9} b^{5}

Simplify

\frac{14 !}{6 ! ( 14 - 6 ) !} a^{8} b^{6} : 3003 a^{8} b^{6}

Simplify

\frac{14 !}{7 ! ( 14 - 7 ) !} a^{7} b^{7} : 3432 a^{7} b^{7}

Simplify

\frac{14 !}{8 ! ( 14 - 8 ) !} a^{6} b^{8} : 3003 a^{6} b^{8}

Simplify

\frac{14 !}{9 ! ( 14 - 9 ) !} a^{5} b^{9} : 2002 a^{5} b^{9}

Simplify

\frac{14 !}{10 ! ( 14 - 10 ) !} a^{4} b^{10} : 1001 a^{4} b^{10}

Simplify

\frac{14 !}{11 ! ( 14 - 11 ) !} a^{3} b^{11} : 364 a^{3} b^{11}

Simplify

\frac{14 !}{12 ! ( 14 - 12 ) !} a^{2} b^{12} : 91 a^{2} b^{12}

Simplify

\frac{14 !}{13 ! ( 14 - 13 ) !} a^{1} b^{13} : 14 a b^{13}

Simplify

\frac{14 !}{14 ! ( 14 - 14 ) !} a^{0} b^{14} : b^{14}

= a^{14} + 14 a^{13} b + 91 a^{12} b^{2} + 364 a^{11} b^{3} + 1001 a^{10} b^{4} + 2002 a^{9} b^{5} + 3003 a^{8} b^{6} + 3432 a^{7} b^{7} + 3003 a^{6} b^{8} + 2002 a^{5} b^{9} + 1001 a^{4} b^{10} + 364 a^{3} b^{11} + 91 a^{2} b^{12} + 14 a b^{13} + b^{14}

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