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Popular Algebra >

expand (x+6)^{20}

Solution

expand

(x + 6)^{20}

Solution

x^{20} + 120 x^{19} + 6840 x^{18} + 246240 x^{17} + 6279120 x^{16} + 120559104 x^{15} + 1808386560 x^{14} + 21700638720 x^{13} + 211581227520 x^{12} + 1692649820160 x^{11} + 1.11715 E 13 x^{10} + 6.09354 E 13 x^{9} + 6^{12} \cdot 125970 x^{8} + 6^{13} \cdot 77520 x^{7} + 6^{14} \cdot 38760 x^{6} + 6^{15} \cdot 15504 x^{5} + 6^{16} \cdot 4845 x^{4} + 6^{17} \cdot 1140 x^{3} + 6^{18} \cdot 190 x^{2} + 6^{19} \cdot 20 x + 6^{20}

Solution steps

(x + 6)^{20}

Apply binomial theorem:

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

a = x, b = 6

= i = 0 \sum 20 (i 20) x^{(20 - i)} \cdot 6^{i}

Expand summation

= \frac{20 !}{0 ! ( 20 - 0 ) !} x^{20} \cdot 6^{0} + \frac{20 !}{1 ! ( 20 - 1 ) !} x^{19} \cdot 6^{1} + \frac{20 !}{2 ! ( 20 - 2 ) !} x^{18} \cdot 6^{2} + \frac{20 !}{3 ! ( 20 - 3 ) !} x^{17} \cdot 6^{3} + \frac{20 !}{4 ! ( 20 - 4 ) !} x^{16} \cdot 6^{4} + \frac{20 !}{5 ! ( 20 - 5 ) !} x^{15} \cdot 6^{5} + \frac{20 !}{6 ! ( 20 - 6 ) !} x^{14} \cdot 6^{6} + \frac{20 !}{7 ! ( 20 - 7 ) !} x^{13} \cdot 6^{7} + \frac{20 !}{8 ! ( 20 - 8 ) !} x^{12} \cdot 6^{8} + \frac{20 !}{9 ! ( 20 - 9 ) !} x^{11} \cdot 6^{9} + \frac{20 !}{10 ! ( 20 - 10 ) !} x^{10} \cdot 6^{10} + \frac{20 !}{11 ! ( 20 - 11 ) !} x^{9} \cdot 6^{11} + \frac{20 !}{12 ! ( 20 - 12 ) !} x^{8} \cdot 6^{12} + \frac{20 !}{13 ! ( 20 - 13 ) !} x^{7} \cdot 6^{13} + \frac{20 !}{14 ! ( 20 - 14 ) !} x^{6} \cdot 6^{14} + \frac{20 !}{15 ! ( 20 - 15 ) !} x^{5} \cdot 6^{15} + \frac{20 !}{16 ! ( 20 - 16 ) !} x^{4} \cdot 6^{16} + \frac{20 !}{17 ! ( 20 - 17 ) !} x^{3} \cdot 6^{17} + \frac{20 !}{18 ! ( 20 - 18 ) !} x^{2} \cdot 6^{18} + \frac{20 !}{19 ! ( 20 - 19 ) !} x^{1} \cdot 6^{19} + \frac{20 !}{20 ! ( 20 - 20 ) !} x^{0} \cdot 6^{20}

Simplify

\frac{20 !}{0 ! ( 20 - 0 ) !} x^{20} \cdot 6^{0} : x^{20}

Simplify

\frac{20 !}{1 ! ( 20 - 1 ) !} x^{19} \cdot 6^{1} : 120 x^{19}

Simplify

\frac{20 !}{2 ! ( 20 - 2 ) !} x^{18} \cdot 6^{2} : 6840 x^{18}

Simplify

\frac{20 !}{3 ! ( 20 - 3 ) !} x^{17} \cdot 6^{3} : 246240 x^{17}

Simplify

\frac{20 !}{4 ! ( 20 - 4 ) !} x^{16} \cdot 6^{4} : 6279120 x^{16}

Simplify

\frac{20 !}{5 ! ( 20 - 5 ) !} x^{15} \cdot 6^{5} : 120559104 x^{15}

Simplify

\frac{20 !}{6 ! ( 20 - 6 ) !} x^{14} \cdot 6^{6} : 1808386560 x^{14}

Simplify

\frac{20 !}{7 ! ( 20 - 7 ) !} x^{13} \cdot 6^{7} : 21700638720 x^{13}

Simplify

\frac{20 !}{8 ! ( 20 - 8 ) !} x^{12} \cdot 6^{8} : 211581227520 x^{12}

Simplify

\frac{20 !}{9 ! ( 20 - 9 ) !} x^{11} \cdot 6^{9} : 1692649820160 x^{11}

Simplify

\frac{20 !}{10 ! ( 20 - 10 ) !} x^{10} \cdot 6^{10} : \frac{4.05391 E 19 x ^{10}}{3628800}

Simplify

\frac{20 !}{11 ! ( 20 - 11 ) !} x^{9} \cdot 6^{11} : \frac{2.21122 E 19 x ^{9}}{362880}

Simplify

\frac{20 !}{12 ! ( 20 - 12 ) !} x^{8} \cdot 6^{12} : 6^{12} \cdot 125970 x^{8}

Simplify

\frac{20 !}{13 ! ( 20 - 13 ) !} x^{7} \cdot 6^{13} : 6^{13} \cdot 77520 x^{7}

Simplify

\frac{20 !}{14 ! ( 20 - 14 ) !} x^{6} \cdot 6^{14} : 6^{14} \cdot 38760 x^{6}

Simplify

\frac{20 !}{15 ! ( 20 - 15 ) !} x^{5} \cdot 6^{15} : 6^{15} \cdot 15504 x^{5}

Simplify

\frac{20 !}{16 ! ( 20 - 16 ) !} x^{4} \cdot 6^{16} : 6^{16} \cdot 4845 x^{4}

Simplify

\frac{20 !}{17 ! ( 20 - 17 ) !} x^{3} \cdot 6^{17} : 6^{17} \cdot 1140 x^{3}

Simplify

\frac{20 !}{18 ! ( 20 - 18 ) !} x^{2} \cdot 6^{18} : 6^{18} \cdot 190 x^{2}

Simplify

\frac{20 !}{19 ! ( 20 - 19 ) !} x^{1} \cdot 6^{19} : 6^{19} \cdot 20 x

Simplify

\frac{20 !}{20 ! ( 20 - 20 ) !} x^{0} \cdot 6^{20} : 6^{20}

= x^{20} + 120 x^{19} + 6840 x^{18} + 246240 x^{17} + 6279120 x^{16} + 120559104 x^{15} + 1808386560 x^{14} + 21700638720 x^{13} + 211581227520 x^{12} + 1692649820160 x^{11} + \frac{4.05391 E 19 x ^{10}}{3628800} + \frac{2.21122 E 19 x ^{9}}{362880} + 6^{12} \cdot 125970 x^{8} + 6^{13} \cdot 77520 x^{7} + 6^{14} \cdot 38760 x^{6} + 6^{15} \cdot 15504 x^{5} + 6^{16} \cdot 4845 x^{4} + 6^{17} \cdot 1140 x^{3} + 6^{18} \cdot 190 x^{2} + 6^{19} \cdot 20 x + 6^{20}

Simplify

x^{20} + 120 x^{19} + 6840 x^{18} + 246240 x^{17} + 6279120 x^{16} + 120559104 x^{15} + 1808386560 x^{14} + 21700638720 x^{13} + 211581227520 x^{12} + 1692649820160 x^{11} + \frac{4.05391 E 19 x ^{10}}{3628800} + \frac{2.21122 E 19 x ^{9}}{362880} + 6^{12} \cdot 125970 x^{8} + 6^{13} \cdot 77520 x^{7} + 6^{14} \cdot 38760 x^{6} + 6^{15} \cdot 15504 x^{5} + 6^{16} \cdot 4845 x^{4} + 6^{17} \cdot 1140 x^{3} + 6^{18} \cdot 190 x^{2} + 6^{19} \cdot 20 x + 6^{20} : x^{20} + 120 x^{19} + 6840 x^{18} + 246240 x^{17} + 6279120 x^{16} + 120559104 x^{15} + 1808386560 x^{14} + 21700638720 x^{13} + 211581227520 x^{12} + 1692649820160 x^{11} + 1.11715 E 13 x^{10} + 6.09354 E 13 x^{9} + 6^{12} \cdot 125970 x^{8} + 6^{13} \cdot 77520 x^{7} + 6^{14} \cdot 38760 x^{6} + 6^{15} \cdot 15504 x^{5} + 6^{16} \cdot 4845 x^{4} + 6^{17} \cdot 1140 x^{3} + 6^{18} \cdot 190 x^{2} + 6^{19} \cdot 20 x + 6^{20}

= x^{20} + 120 x^{19} + 6840 x^{18} + 246240 x^{17} + 6279120 x^{16} + 120559104 x^{15} + 1808386560 x^{14} + 21700638720 x^{13} + 211581227520 x^{12} + 1692649820160 x^{11} + 1.11715 E 13 x^{10} + 6.09354 E 13 x^{9} + 6^{12} \cdot 125970 x^{8} + 6^{13} \cdot 77520 x^{7} + 6^{14} \cdot 38760 x^{6} + 6^{15} \cdot 15504 x^{5} + 6^{16} \cdot 4845 x^{4} + 6^{17} \cdot 1140 x^{3} + 6^{18} \cdot 190 x^{2} + 6^{19} \cdot 20 x + 6^{20}

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