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Popular Algebra >

expand (2x^3+1/x)^{29}

Solution

expand

(2 x^{3} + \frac{1}{x})^{29}

Solution

536870912 x^{87} + 7784628224 x^{83} + 54492397568 x^{79} + 245215789056 x^{75} + 796951314432 x^{71} + 1992378286080 x^{67} + 3984756572160 x^{63} + 6546385797120 x^{59} + 9001280471040 x^{55} + 1.05015 E 13 x^{51} + 1.05015 E 13 x^{47} + \frac{5.17178 E 19 x ^{43}}{5702400} + \frac{4.6546 E 20 x ^{39}}{68428800} + \frac{3.95641 E 21 x ^{35}}{889574400} + \frac{4.52161 E 21 x ^{31}}{1779148800} + \frac{2.2608 E 21 x ^{27}}{1779148800} + \frac{4.94551 E 20 x ^{23}}{889574400} + \frac{1.45456 E 19 x ^{19}}{68428800} + 70855249920 x^{15} + 20510730240 x^{11} + 5127682560 x^{7} + 1098789120 x^{3} + \frac{199779840}{x} + \frac{30401280}{x ^{5}} + \frac{3800160}{x ^{9}} + \frac{380016}{x ^{13}} + \frac{29232}{x ^{17}} + \frac{1624}{x ^{21}} + \frac{58}{x ^{25}} + \frac{1}{x ^{29}}

Solution steps

(2 x^{3} + \frac{1}{x})^{29}

Apply binomial theorem:

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

a = 2 x^{3}, b = \frac{1}{x}

= i = 0 \sum 29 (i 29) (2 x^{3})^{(29 - i)} (\frac{1}{x})^{i}

Expand summation

= \frac{29 !}{0 ! ( 29 - 0 ) !} (2 x^{3})^{29} (\frac{1}{x})^{0} + \frac{29 !}{1 ! ( 29 - 1 ) !} (2 x^{3})^{28} (\frac{1}{x})^{1} + \frac{29 !}{2 ! ( 29 - 2 ) !} (2 x^{3})^{27} (\frac{1}{x})^{2} + \frac{29 !}{3 ! ( 29 - 3 ) !} (2 x^{3})^{26} (\frac{1}{x})^{3} + \frac{29 !}{4 ! ( 29 - 4 ) !} (2 x^{3})^{25} (\frac{1}{x})^{4} + \frac{29 !}{5 ! ( 29 - 5 ) !} (2 x^{3})^{24} (\frac{1}{x})^{5} + \frac{29 !}{6 ! ( 29 - 6 ) !} (2 x^{3})^{23} (\frac{1}{x})^{6} + \frac{29 !}{7 ! ( 29 - 7 ) !} (2 x^{3})^{22} (\frac{1}{x})^{7} + \frac{29 !}{8 ! ( 29 - 8 ) !} (2 x^{3})^{21} (\frac{1}{x})^{8} + \frac{29 !}{9 ! ( 29 - 9 ) !} (2 x^{3})^{20} (\frac{1}{x})^{9} + \frac{29 !}{10 ! ( 29 - 10 ) !} (2 x^{3})^{19} (\frac{1}{x})^{10} + \frac{29 !}{11 ! ( 29 - 11 ) !} (2 x^{3})^{18} (\frac{1}{x})^{11} + \frac{29 !}{12 ! ( 29 - 12 ) !} (2 x^{3})^{17} (\frac{1}{x})^{12} + \frac{29 !}{13 ! ( 29 - 13 ) !} (2 x^{3})^{16} (\frac{1}{x})^{13} + \frac{29 !}{14 ! ( 29 - 14 ) !} (2 x^{3})^{15} (\frac{1}{x})^{14} + \frac{29 !}{15 ! ( 29 - 15 ) !} (2 x^{3})^{14} (\frac{1}{x})^{15} + \frac{29 !}{16 ! ( 29 - 16 ) !} (2 x^{3})^{13} (\frac{1}{x})^{16} + \frac{29 !}{17 ! ( 29 - 17 ) !} (2 x^{3})^{12} (\frac{1}{x})^{17} + \frac{29 !}{18 ! ( 29 - 18 ) !} (2 x^{3})^{11} (\frac{1}{x})^{18} + \frac{29 !}{19 ! ( 29 - 19 ) !} (2 x^{3})^{10} (\frac{1}{x})^{19} + \frac{29 !}{20 ! ( 29 - 20 ) !} (2 x^{3})^{9} (\frac{1}{x})^{20} + \frac{29 !}{21 ! ( 29 - 21 ) !} (2 x^{3})^{8} (\frac{1}{x})^{21} + \frac{29 !}{22 ! ( 29 - 22 ) !} (2 x^{3})^{7} (\frac{1}{x})^{22} + \frac{29 !}{23 ! ( 29 - 23 ) !} (2 x^{3})^{6} (\frac{1}{x})^{23} + \frac{29 !}{24 ! ( 29 - 24 ) !} (2 x^{3})^{5} (\frac{1}{x})^{24} + \frac{29 !}{25 ! ( 29 - 25 ) !} (2 x^{3})^{4} (\frac{1}{x})^{25} + \frac{29 !}{26 ! ( 29 - 26 ) !} (2 x^{3})^{3} (\frac{1}{x})^{26} + \frac{29 !}{27 ! ( 29 - 27 ) !} (2 x^{3})^{2} (\frac{1}{x})^{27} + \frac{29 !}{28 ! ( 29 - 28 ) !} (2 x^{3})^{1} (\frac{1}{x})^{28} + \frac{29 !}{29 ! ( 29 - 29 ) !} (2 x^{3})^{0} (\frac{1}{x})^{29}

Simplify

\frac{29 !}{0 ! ( 29 - 0 ) !} (2 x^{3})^{29} (\frac{1}{x})^{0} : 536870912 x^{87}

Simplify

\frac{29 !}{1 ! ( 29 - 1 ) !} (2 x^{3})^{28} (\frac{1}{x})^{1} : 7784628224 x^{83}

Simplify

\frac{29 !}{2 ! ( 29 - 2 ) !} (2 x^{3})^{27} (\frac{1}{x})^{2} : 54492397568 x^{79}

Simplify

\frac{29 !}{3 ! ( 29 - 3 ) !} (2 x^{3})^{26} (\frac{1}{x})^{3} : 245215789056 x^{75}

Simplify

\frac{29 !}{4 ! ( 29 - 4 ) !} (2 x^{3})^{25} (\frac{1}{x})^{4} : 796951314432 x^{71}

Simplify

\frac{29 !}{5 ! ( 29 - 5 ) !} (2 x^{3})^{24} (\frac{1}{x})^{5} : 1992378286080 x^{67}

Simplify

\frac{29 !}{6 ! ( 29 - 6 ) !} (2 x^{3})^{23} (\frac{1}{x})^{6} : 3984756572160 x^{63}

Simplify

\frac{29 !}{7 ! ( 29 - 7 ) !} (2 x^{3})^{22} (\frac{1}{x})^{7} : 6546385797120 x^{59}

Simplify

\frac{29 !}{8 ! ( 29 - 8 ) !} (2 x^{3})^{21} (\frac{1}{x})^{8} : 9001280471040 x^{55}

Simplify

\frac{29 !}{9 ! ( 29 - 9 ) !} (2 x^{3})^{20} (\frac{1}{x})^{9} : 1.05015 E 13 x^{51}

Simplify

\frac{29 !}{10 ! ( 29 - 10 ) !} (2 x^{3})^{19} (\frac{1}{x})^{10} : \frac{3.81078 E 19 x ^{47}}{3628800}

Simplify

\frac{29 !}{11 ! ( 29 - 11 ) !} (2 x^{3})^{18} (\frac{1}{x})^{11} : \frac{3.62024 E 20 x ^{43}}{39916800}

Simplify

\frac{29 !}{12 ! ( 29 - 12 ) !} (2 x^{3})^{17} (\frac{1}{x})^{12} : \frac{3.25822 E 21 x ^{39}}{479001600}

Simplify

\frac{29 !}{13 ! ( 29 - 13 ) !} (2 x^{3})^{16} (\frac{1}{x})^{13} : \frac{2.76949 E 22 x ^{35}}{6227020800}

Simplify

\frac{29 !}{14 ! ( 29 - 14 ) !} (2 x^{3})^{15} (\frac{1}{x})^{14} : \frac{2.21559 E 23 x ^{31}}{87178291200}

Simplify

\frac{29 !}{15 ! ( 29 - 15 ) !} (2 x^{3})^{14} (\frac{1}{x})^{15} : \frac{1.10779 E 23 x ^{27}}{87178291200}

Simplify

\frac{29 !}{16 ! ( 29 - 16 ) !} (2 x^{3})^{13} (\frac{1}{x})^{16} : \frac{3.46186 E 21 x ^{23}}{6227020800}

Simplify

\frac{29 !}{17 ! ( 29 - 17 ) !} (2 x^{3})^{12} (\frac{1}{x})^{17} : \frac{1.01819 E 20 x ^{19}}{479001600}

Simplify

\frac{29 !}{18 ! ( 29 - 18 ) !} (2 x^{3})^{11} (\frac{1}{x})^{18} : 70855249920 x^{15}

Simplify

\frac{29 !}{19 ! ( 29 - 19 ) !} (2 x^{3})^{10} (\frac{1}{x})^{19} : 20510730240 x^{11}

Simplify

\frac{29 !}{20 ! ( 29 - 20 ) !} (2 x^{3})^{9} (\frac{1}{x})^{20} : 5127682560 x^{7}

Simplify

\frac{29 !}{21 ! ( 29 - 21 ) !} (2 x^{3})^{8} (\frac{1}{x})^{21} : 1098789120 x^{3}

Simplify

\frac{29 !}{22 ! ( 29 - 22 ) !} (2 x^{3})^{7} (\frac{1}{x})^{22} : \frac{199779840}{x}

Simplify

\frac{29 !}{23 ! ( 29 - 23 ) !} (2 x^{3})^{6} (\frac{1}{x})^{23} : \frac{30401280}{x ^{5}}

Simplify

\frac{29 !}{24 ! ( 29 - 24 ) !} (2 x^{3})^{5} (\frac{1}{x})^{24} : \frac{3800160}{x ^{9}}

Simplify

\frac{29 !}{25 ! ( 29 - 25 ) !} (2 x^{3})^{4} (\frac{1}{x})^{25} : \frac{380016}{x ^{13}}

Simplify

\frac{29 !}{26 ! ( 29 - 26 ) !} (2 x^{3})^{3} (\frac{1}{x})^{26} : \frac{29232}{x ^{17}}

Simplify

\frac{29 !}{27 ! ( 29 - 27 ) !} (2 x^{3})^{2} (\frac{1}{x})^{27} : \frac{1624}{x ^{21}}

Simplify

\frac{29 !}{28 ! ( 29 - 28 ) !} (2 x^{3})^{1} (\frac{1}{x})^{28} : \frac{58}{x ^{25}}

Simplify

\frac{29 !}{29 ! ( 29 - 29 ) !} (2 x^{3})^{0} (\frac{1}{x})^{29} : \frac{1}{x ^{29}}

= 536870912 x^{87} + 7784628224 x^{83} + 54492397568 x^{79} + 245215789056 x^{75} + 796951314432 x^{71} + 1992378286080 x^{67} + 3984756572160 x^{63} + 6546385797120 x^{59} + 9001280471040 x^{55} + 1.05015 E 13 x^{51} + \frac{3.81078 E 19 x ^{47}}{3628800} + \frac{3.62024 E 20 x ^{43}}{39916800} + \frac{3.25822 E 21 x ^{39}}{479001600} + \frac{2.76949 E 22 x ^{35}}{6227020800} + \frac{2.21559 E 23 x ^{31}}{87178291200} + \frac{1.10779 E 23 x ^{27}}{87178291200} + \frac{3.46186 E 21 x ^{23}}{6227020800} + \frac{1.01819 E 20 x ^{19}}{479001600} + 70855249920 x^{15} + 20510730240 x^{11} + 5127682560 x^{7} + 1098789120 x^{3} + \frac{199779840}{x} + \frac{30401280}{x ^{5}} + \frac{3800160}{x ^{9}} + \frac{380016}{x ^{13}} + \frac{29232}{x ^{17}} + \frac{1624}{x ^{21}} + \frac{58}{x ^{25}} + \frac{1}{x ^{29}}

\frac{3.81078 E 19 x ^{47}}{3628800} = 1.05015 E 13 x^{47}

\frac{3.62024 E 20 x ^{43}}{39916800} = \frac{5.17178 E 19 x ^{43}}{5702400}

\frac{3.25822 E 21 x ^{39}}{479001600} = \frac{4.6546 E 20 x ^{39}}{68428800}

\frac{2.76949 E 22 x ^{35}}{6227020800} = \frac{3.95641 E 21 x ^{35}}{889574400}

\frac{2.21559 E 23 x ^{31}}{87178291200} = \frac{4.52161 E 21 x ^{31}}{1779148800}

\frac{1.10779 E 23 x ^{27}}{87178291200} = \frac{2.2608 E 21 x ^{27}}{1779148800}

\frac{3.46186 E 21 x ^{23}}{6227020800} = \frac{4.94551 E 20 x ^{23}}{889574400}

\frac{1.01819 E 20 x ^{19}}{479001600} = \frac{1.45456 E 19 x ^{19}}{68428800}

= 536870912 x^{87} + 7784628224 x^{83} + 54492397568 x^{79} + 245215789056 x^{75} + 796951314432 x^{71} + 1992378286080 x^{67} + 3984756572160 x^{63} + 6546385797120 x^{59} + 9001280471040 x^{55} + 1.05015 E 13 x^{51} + 1.05015 E 13 x^{47} + \frac{5.17178 E 19 x ^{43}}{5702400} + \frac{4.6546 E 20 x ^{39}}{68428800} + \frac{3.95641 E 21 x ^{35}}{889574400} + \frac{4.52161 E 21 x ^{31}}{1779148800} + \frac{2.2608 E 21 x ^{27}}{1779148800} + \frac{4.94551 E 20 x ^{23}}{889574400} + \frac{1.45456 E 19 x ^{19}}{68428800} + 70855249920 x^{15} + 20510730240 x^{11} + 5127682560 x^{7} + 1098789120 x^{3} + \frac{199779840}{x} + \frac{30401280}{x ^{5}} + \frac{3800160}{x ^{9}} + \frac{380016}{x ^{13}} + \frac{29232}{x ^{17}} + \frac{1624}{x ^{21}} + \frac{58}{x ^{25}} + \frac{1}{x ^{29}}

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