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Popular Algebra >

expand (a+b)^{13}

Solution

expand

(a + b)^{13}

Solution

a^{13} + 13 a^{12} b + 78 a^{11} b^{2} + 286 a^{10} b^{3} + 715 a^{9} b^{4} + 1287 a^{8} b^{5} + 1716 a^{7} b^{6} + 1716 a^{6} b^{7} + 1287 a^{5} b^{8} + 715 a^{4} b^{9} + 286 a^{3} b^{10} + 78 a^{2} b^{11} + 13 a b^{12} + b^{13}

Solution steps

(a + b)^{13}

Apply binomial theorem:

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

a = a, b = b

= i = 0 \sum 13 (i 13) a^{(13 - i)} b^{i}

Expand summation

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

Simplify

\frac{13 !}{0 ! ( 13 - 0 ) !} a^{13} b^{0} : a^{13}

Simplify

\frac{13 !}{1 ! ( 13 - 1 ) !} a^{12} b^{1} : 13 a^{12} b

Simplify

\frac{13 !}{2 ! ( 13 - 2 ) !} a^{11} b^{2} : 78 a^{11} b^{2}

Simplify

\frac{13 !}{3 ! ( 13 - 3 ) !} a^{10} b^{3} : 286 a^{10} b^{3}

Simplify

\frac{13 !}{4 ! ( 13 - 4 ) !} a^{9} b^{4} : 715 a^{9} b^{4}

Simplify

\frac{13 !}{5 ! ( 13 - 5 ) !} a^{8} b^{5} : 1287 a^{8} b^{5}

Simplify

\frac{13 !}{6 ! ( 13 - 6 ) !} a^{7} b^{6} : 1716 a^{7} b^{6}

Simplify

\frac{13 !}{7 ! ( 13 - 7 ) !} a^{6} b^{7} : 1716 a^{6} b^{7}

Simplify

\frac{13 !}{8 ! ( 13 - 8 ) !} a^{5} b^{8} : 1287 a^{5} b^{8}

Simplify

\frac{13 !}{9 ! ( 13 - 9 ) !} a^{4} b^{9} : 715 a^{4} b^{9}

Simplify

\frac{13 !}{10 ! ( 13 - 10 ) !} a^{3} b^{10} : 286 a^{3} b^{10}

Simplify

\frac{13 !}{11 ! ( 13 - 11 ) !} a^{2} b^{11} : 78 a^{2} b^{11}

Simplify

\frac{13 !}{12 ! ( 13 - 12 ) !} a^{1} b^{12} : 13 a b^{12}

Simplify

\frac{13 !}{13 ! ( 13 - 13 ) !} a^{0} b^{13} : b^{13}

= a^{13} + 13 a^{12} b + 78 a^{11} b^{2} + 286 a^{10} b^{3} + 715 a^{9} b^{4} + 1287 a^{8} b^{5} + 1716 a^{7} b^{6} + 1716 a^{6} b^{7} + 1287 a^{5} b^{8} + 715 a^{4} b^{9} + 286 a^{3} b^{10} + 78 a^{2} b^{11} + 13 a b^{12} + b^{13}

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Frequently Asked Questions (FAQ)

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