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Popolare Algebra >

espandere (5-\sqrt[4]{x/(1220703125)})^{19}

Soluzione

espandere

(5 - 4 \frac{x}{1220703125})^{19}

Soluzione

5^{19} - 4 5^{59} \cdot 19 4 x + 5^{17} \cdot 171 (4 \frac{x}{1220703125})^{2} - 5^{16} \cdot 969 (4 \frac{x}{1220703125})^{3} + 96900 x - 58140 x 4 \frac{x}{1220703125} + \frac{27132 x ^{3}}{1220703125} - \frac{50388 x 4 x ^{3}}{5 4 122070312 5 ^{3}} + \frac{75582 x ^{2}}{30517578125} - \frac{92378 x ^{2} 4 \frac{x}{1220703125}}{152587890625} + \frac{92378 x ^{5}}{1953125 \cdot 62 5 ^{2} 1220703125} - \frac{75582 x ^{2} 4 x ^{3}}{390625 \cdot 312 5 ^{2} 4 122070312 5 ^{3}} + \frac{50388 x ^{3}}{5 ^{32}} - \frac{27132 x ^{3} 4 \frac{x}{1220703125}}{1562 5 ^{5} \cdot 125} + \frac{11628 x ^{7}}{312 5 ^{5} \cdot 12 5 ^{3} 1220703125} - \frac{3876 x ^{3} 4 x ^{3}}{62 5 ^{8} \cdot 125 4 122070312 5 ^{3}} + \frac{969 x ^{4}}{5 ^{49}} - \frac{171 x ^{4} 4 \frac{x}{1220703125}}{2 5 ^{23} \cdot 625} + \frac{19 x ^{9}}{5 ^{51} 1220703125} - \frac{x ^{4} ( 4 \frac{x}{1220703125} ) ^{3}}{122070312 5 ^{4}}

Fasi della soluzione

(5 - 4 \frac{x}{1220703125})^{19}

Applicare il teorema binomiale:

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

a = 5, b = - 4 \frac{x}{1220703125}

= i = 0 \sum 19 (i 19) \cdot 5^{(19 - i)} (- 4 \frac{x}{1220703125})^{i}

Espandere la sommatoria

= \frac{19 !}{0 ! ( 19 - 0 ) !} \cdot 5^{19} (- 4 \frac{x}{1220703125})^{0} + \frac{19 !}{1 ! ( 19 - 1 ) !} \cdot 5^{18} (- 4 \frac{x}{1220703125})^{1} + \frac{19 !}{2 ! ( 19 - 2 ) !} \cdot 5^{17} (- 4 \frac{x}{1220703125})^{2} + \frac{19 !}{3 ! ( 19 - 3 ) !} \cdot 5^{16} (- 4 \frac{x}{1220703125})^{3} + \frac{19 !}{4 ! ( 19 - 4 ) !} \cdot 5^{15} (- 4 \frac{x}{1220703125})^{4} + \frac{19 !}{5 ! ( 19 - 5 ) !} \cdot 5^{14} (- 4 \frac{x}{1220703125})^{5} + \frac{19 !}{6 ! ( 19 - 6 ) !} \cdot 5^{13} (- 4 \frac{x}{1220703125})^{6} + \frac{19 !}{7 ! ( 19 - 7 ) !} \cdot 5^{12} (- 4 \frac{x}{1220703125})^{7} + \frac{19 !}{8 ! ( 19 - 8 ) !} \cdot 5^{11} (- 4 \frac{x}{1220703125})^{8} + \frac{19 !}{9 ! ( 19 - 9 ) !} \cdot 5^{10} (- 4 \frac{x}{1220703125})^{9} + \frac{19 !}{10 ! ( 19 - 10 ) !} \cdot 5^{9} (- 4 \frac{x}{1220703125})^{10} + \frac{19 !}{11 ! ( 19 - 11 ) !} \cdot 5^{8} (- 4 \frac{x}{1220703125})^{11} + \frac{19 !}{12 ! ( 19 - 12 ) !} \cdot 5^{7} (- 4 \frac{x}{1220703125})^{12} + \frac{19 !}{13 ! ( 19 - 13 ) !} \cdot 5^{6} (- 4 \frac{x}{1220703125})^{13} + \frac{19 !}{14 ! ( 19 - 14 ) !} \cdot 5^{5} (- 4 \frac{x}{1220703125})^{14} + \frac{19 !}{15 ! ( 19 - 15 ) !} \cdot 5^{4} (- 4 \frac{x}{1220703125})^{15} + \frac{19 !}{16 ! ( 19 - 16 ) !} \cdot 5^{3} (- 4 \frac{x}{1220703125})^{16} + \frac{19 !}{17 ! ( 19 - 17 ) !} \cdot 5^{2} (- 4 \frac{x}{1220703125})^{17} + \frac{19 !}{18 ! ( 19 - 18 ) !} \cdot 5^{1} (- 4 \frac{x}{1220703125})^{18} + \frac{19 !}{19 ! ( 19 - 19 ) !} \cdot 5^{0} (- 4 \frac{x}{1220703125})^{19}

Semplificare

\frac{19 !}{0 ! ( 19 - 0 ) !} \cdot 5^{19} (- 4 \frac{x}{1220703125})^{0} : 5^{19}

Semplificare

\frac{19 !}{1 ! ( 19 - 1 ) !} \cdot 5^{18} (- 4 \frac{x}{1220703125})^{1} : - \frac{5 ^{18} \cdot 19 4 x}{4 1220703125}

Semplificare

\frac{19 !}{2 ! ( 19 - 2 ) !} \cdot 5^{17} (- 4 \frac{x}{1220703125})^{2} : 5^{17} \cdot 171 (4 \frac{x}{1220703125})^{2}

Semplificare

\frac{19 !}{3 ! ( 19 - 3 ) !} \cdot 5^{16} (- 4 \frac{x}{1220703125})^{3} : - 5^{16} \cdot 969 (4 \frac{x}{1220703125})^{3}

Semplificare

\frac{19 !}{4 ! ( 19 - 4 ) !} \cdot 5^{15} (- 4 \frac{x}{1220703125})^{4} : \frac{5 ^{15} \cdot 3876 x}{1220703125}

Semplificare

\frac{19 !}{5 ! ( 19 - 5 ) !} \cdot 5^{14} (- 4 \frac{x}{1220703125})^{5} : - 5^{14} \cdot 11628 (4 \frac{x}{1220703125})^{5}

Semplificare

\frac{19 !}{6 ! ( 19 - 6 ) !} \cdot 5^{13} (- 4 \frac{x}{1220703125})^{6} : 5^{13} \cdot 27132 (4 \frac{x}{1220703125})^{6}

Semplificare

\frac{19 !}{7 ! ( 19 - 7 ) !} \cdot 5^{12} (- 4 \frac{x}{1220703125})^{7} : - 1.23018 E 13 (4 \frac{x}{1220703125})^{7}

Semplificare

\frac{19 !}{8 ! ( 19 - 8 ) !} \cdot 5^{11} (- 4 \frac{x}{1220703125})^{8} : \frac{3690527343750 x ^{2}}{122070312 5 ^{2}}

Semplificare

\frac{19 !}{9 ! ( 19 - 9 ) !} \cdot 5^{10} (- 4 \frac{x}{1220703125})^{9} : - 902128906250 (4 \frac{x}{1220703125})^{9}

Semplificare

\frac{19 !}{10 ! ( 19 - 10 ) !} \cdot 5^{9} (- 4 \frac{x}{1220703125})^{10} : 180425781250 (4 \frac{x}{1220703125})^{10}

Semplificare

\frac{19 !}{11 ! ( 19 - 11 ) !} \cdot 5^{8} (- 4 \frac{x}{1220703125})^{11} : - 29524218750 (4 \frac{x}{1220703125})^{11}

Semplificare

\frac{19 !}{12 ! ( 19 - 12 ) !} \cdot 5^{7} (- 4 \frac{x}{1220703125})^{12} : \frac{3936562500 x ^{3}}{122070312 5 ^{3}}

Semplificare

\frac{19 !}{13 ! ( 19 - 13 ) !} \cdot 5^{6} (- 4 \frac{x}{1220703125})^{13} : - 423937500 (4 \frac{x}{1220703125})^{13}

Semplificare

\frac{19 !}{14 ! ( 19 - 14 ) !} \cdot 5^{5} (- 4 \frac{x}{1220703125})^{14} : 36337500 (4 \frac{x}{1220703125})^{14}

Semplificare

\frac{19 !}{15 ! ( 19 - 15 ) !} \cdot 5^{4} (- 4 \frac{x}{1220703125})^{15} : - 2422500 (4 \frac{x}{1220703125})^{15}

Semplificare

\frac{19 !}{16 ! ( 19 - 16 ) !} \cdot 5^{3} (- 4 \frac{x}{1220703125})^{16} : \frac{121125 x ^{4}}{122070312 5 ^{4}}

Semplificare

\frac{19 !}{17 ! ( 19 - 17 ) !} \cdot 5^{2} (- 4 \frac{x}{1220703125})^{17} : - 4275 (4 \frac{x}{1220703125})^{17}

Semplificare

\frac{19 !}{18 ! ( 19 - 18 ) !} \cdot 5^{1} (- 4 \frac{x}{1220703125})^{18} : 95 (4 \frac{x}{1220703125})^{18}

Semplificare

\frac{19 !}{19 ! ( 19 - 19 ) !} \cdot 5^{0} (- 4 \frac{x}{1220703125})^{19} : - (4 \frac{x}{1220703125})^{19}

= 5^{19} - \frac{5 ^{18} \cdot 19 4 x}{4 1220703125} + 5^{17} \cdot 171 (4 \frac{x}{1220703125})^{2} - 5^{16} \cdot 969 (4 \frac{x}{1220703125})^{3} + \frac{5 ^{15} \cdot 3876 x}{1220703125} - 5^{14} \cdot 11628 (4 \frac{x}{1220703125})^{5} + 5^{13} \cdot 27132 (4 \frac{x}{1220703125})^{6} - 1.23018 E 13 (4 \frac{x}{1220703125})^{7} + \frac{3690527343750 x ^{2}}{122070312 5 ^{2}} - 902128906250 (4 \frac{x}{1220703125})^{9} + 180425781250 (4 \frac{x}{1220703125})^{10} - 29524218750 (4 \frac{x}{1220703125})^{11} + \frac{3936562500 x ^{3}}{122070312 5 ^{3}} - 423937500 (4 \frac{x}{1220703125})^{13} + 36337500 (4 \frac{x}{1220703125})^{14} - 2422500 (4 \frac{x}{1220703125})^{15} + \frac{121125 x ^{4}}{122070312 5 ^{4}} - 4275 (4 \frac{x}{1220703125})^{17} + 95 (4 \frac{x}{1220703125})^{18} - (4 \frac{x}{1220703125})^{19}

Semplificare

5^{19} - \frac{5 ^{18} \cdot 19 4 x}{4 1220703125} + 5^{17} \cdot 171 (4 \frac{x}{1220703125})^{2} - 5^{16} \cdot 969 (4 \frac{x}{1220703125})^{3} + \frac{5 ^{15} \cdot 3876 x}{1220703125} - 5^{14} \cdot 11628 (4 \frac{x}{1220703125})^{5} + 5^{13} \cdot 27132 (4 \frac{x}{1220703125})^{6} - 1.23018 E 13 (4 \frac{x}{1220703125})^{7} + \frac{3690527343750 x ^{2}}{122070312 5 ^{2}} - 902128906250 (4 \frac{x}{1220703125})^{9} + 180425781250 (4 \frac{x}{1220703125})^{10} - 29524218750 (4 \frac{x}{1220703125})^{11} + \frac{3936562500 x ^{3}}{122070312 5 ^{3}} - 423937500 (4 \frac{x}{1220703125})^{13} + 36337500 (4 \frac{x}{1220703125})^{14} - 2422500 (4 \frac{x}{1220703125})^{15} + \frac{121125 x ^{4}}{122070312 5 ^{4}} - 4275 (4 \frac{x}{1220703125})^{17} + 95 (4 \frac{x}{1220703125})^{18} - (4 \frac{x}{1220703125})^{19} : 5^{19} - 5^{\frac{59}{4}} \cdot 19 4 x + 5^{17} \cdot 171 (4 \frac{x}{1220703125})^{2} - 5^{16} \cdot 969 (4 \frac{x}{1220703125})^{3} + 96900 x - 58140 x 4 \frac{x}{1220703125} + \frac{27132 x ^{\frac{3}{2}}}{1220703125} - \frac{50388 x 4 x ^{3}}{5 4 122070312 5 ^{3}} + \frac{75582 x ^{2}}{30517578125} - \frac{92378 x ^{2} 4 \frac{x}{1220703125}}{152587890625} + \frac{92378 x ^{\frac{5}{2}}}{1953125 \cdot 62 5 ^{2} 1220703125} - \frac{75582 x ^{2} 4 x ^{3}}{390625 \cdot 312 5 ^{2} 4 122070312 5 ^{3}} + \frac{50388 x ^{3}}{5 ^{32}} - \frac{27132 x ^{3} 4 \frac{x}{1220703125}}{1562 5 ^{5} \cdot 125} + \frac{11628 x ^{\frac{7}{2}}}{312 5 ^{5} \cdot 12 5 ^{3} 1220703125} - \frac{3876 x ^{3} 4 x ^{3}}{62 5 ^{8} \cdot 125 4 122070312 5 ^{3}} + \frac{969 x ^{4}}{5 ^{49}} - \frac{171 x ^{4} 4 \frac{x}{1220703125}}{2 5 ^{23} \cdot 625} + \frac{19 x ^{\frac{9}{2}}}{5 ^{51} 1220703125} - \frac{x ^{4} ( 4 \frac{x}{1220703125} ) ^{3}}{122070312 5 ^{4}}

= 5^{19} - 5^{\frac{59}{4}} \cdot 19 4 x + 5^{17} \cdot 171 (4 \frac{x}{1220703125})^{2} - 5^{16} \cdot 969 (4 \frac{x}{1220703125})^{3} + 96900 x - 58140 x 4 \frac{x}{1220703125} + \frac{27132 x ^{\frac{3}{2}}}{1220703125} - \frac{50388 x 4 x ^{3}}{5 4 122070312 5 ^{3}} + \frac{75582 x ^{2}}{30517578125} - \frac{92378 x ^{2} 4 \frac{x}{1220703125}}{152587890625} + \frac{92378 x ^{\frac{5}{2}}}{1953125 \cdot 62 5 ^{2} 1220703125} - \frac{75582 x ^{2} 4 x ^{3}}{390625 \cdot 312 5 ^{2} 4 122070312 5 ^{3}} + \frac{50388 x ^{3}}{5 ^{32}} - \frac{27132 x ^{3} 4 \frac{x}{1220703125}}{1562 5 ^{5} \cdot 125} + \frac{11628 x ^{\frac{7}{2}}}{312 5 ^{5} \cdot 12 5 ^{3} 1220703125} - \frac{3876 x ^{3} 4 x ^{3}}{62 5 ^{8} \cdot 125 4 122070312 5 ^{3}} + \frac{969 x ^{4}}{5 ^{49}} - \frac{171 x ^{4} 4 \frac{x}{1220703125}}{2 5 ^{23} \cdot 625} + \frac{19 x ^{\frac{9}{2}}}{5 ^{51} 1220703125} - \frac{x ^{4} ( 4 \frac{x}{1220703125} ) ^{3}}{122070312 5 ^{4}}

Semplificare

5^{19} - 5^{\frac{59}{4}} \cdot 19 4 x + 5^{17} \cdot 171 (4 \frac{x}{1220703125})^{2} - 5^{16} \cdot 969 (4 \frac{x}{1220703125})^{3} + 96900 x - 58140 x 4 \frac{x}{1220703125} + \frac{27132 x ^{\frac{3}{2}}}{1220703125} - \frac{50388 x 4 x ^{3}}{5 4 122070312 5 ^{3}} + \frac{75582 x ^{2}}{30517578125} - \frac{92378 x ^{2} 4 \frac{x}{1220703125}}{152587890625} + \frac{92378 x ^{\frac{5}{2}}}{1953125 \cdot 62 5 ^{2} 1220703125} - \frac{75582 x ^{2} 4 x ^{3}}{390625 \cdot 312 5 ^{2} 4 122070312 5 ^{3}} + \frac{50388 x ^{3}}{5 ^{32}} - \frac{27132 x ^{3} 4 \frac{x}{1220703125}}{1562 5 ^{5} \cdot 125} + \frac{11628 x ^{\frac{7}{2}}}{312 5 ^{5} \cdot 12 5 ^{3} 1220703125} - \frac{3876 x ^{3} 4 x ^{3}}{62 5 ^{8} \cdot 125 4 122070312 5 ^{3}} + \frac{969 x ^{4}}{5 ^{49}} - \frac{171 x ^{4} 4 \frac{x}{1220703125}}{2 5 ^{23} \cdot 625} + \frac{19 x ^{\frac{9}{2}}}{5 ^{51} 1220703125} - \frac{x ^{4} ( 4 \frac{x}{1220703125} ) ^{3}}{122070312 5 ^{4}} : 5^{19} - 4 5^{59} \cdot 19 4 x + 5^{17} \cdot 171 (4 \frac{x}{1220703125})^{2} - 5^{16} \cdot 969 (4 \frac{x}{1220703125})^{3} + 96900 x - 58140 x 4 \frac{x}{1220703125} + \frac{27132 x ^{3}}{1220703125} - \frac{50388 x 4 x ^{3}}{5 4 122070312 5 ^{3}} + \frac{75582 x ^{2}}{30517578125} - \frac{92378 x ^{2} 4 \frac{x}{1220703125}}{152587890625} + \frac{92378 x ^{5}}{1953125 \cdot 62 5 ^{2} 1220703125} - \frac{75582 x ^{2} 4 x ^{3}}{390625 \cdot 312 5 ^{2} 4 122070312 5 ^{3}} + \frac{50388 x ^{3}}{5 ^{32}} - \frac{27132 x ^{3} 4 \frac{x}{1220703125}}{1562 5 ^{5} \cdot 125} + \frac{11628 x ^{7}}{312 5 ^{5} \cdot 12 5 ^{3} 1220703125} - \frac{3876 x ^{3} 4 x ^{3}}{62 5 ^{8} \cdot 125 4 122070312 5 ^{3}} + \frac{969 x ^{4}}{5 ^{49}} - \frac{171 x ^{4} 4 \frac{x}{1220703125}}{2 5 ^{23} \cdot 625} + \frac{19 x ^{9}}{5 ^{51} 1220703125} - \frac{x ^{4} ( 4 \frac{x}{1220703125} ) ^{3}}{122070312 5 ^{4}}

= 5^{19} - 4 5^{59} \cdot 19 4 x + 5^{17} \cdot 171 (4 \frac{x}{1220703125})^{2} - 5^{16} \cdot 969 (4 \frac{x}{1220703125})^{3} + 96900 x - 58140 x 4 \frac{x}{1220703125} + \frac{27132 x ^{3}}{1220703125} - \frac{50388 x 4 x ^{3}}{5 4 122070312 5 ^{3}} + \frac{75582 x ^{2}}{30517578125} - \frac{92378 x ^{2} 4 \frac{x}{1220703125}}{152587890625} + \frac{92378 x ^{5}}{1953125 \cdot 62 5 ^{2} 1220703125} - \frac{75582 x ^{2} 4 x ^{3}}{390625 \cdot 312 5 ^{2} 4 122070312 5 ^{3}} + \frac{50388 x ^{3}}{5 ^{32}} - \frac{27132 x ^{3} 4 \frac{x}{1220703125}}{1562 5 ^{5} \cdot 125} + \frac{11628 x ^{7}}{312 5 ^{5} \cdot 12 5 ^{3} 1220703125} - \frac{3876 x ^{3} 4 x ^{3}}{62 5 ^{8} \cdot 125 4 122070312 5 ^{3}} + \frac{969 x ^{4}}{5 ^{49}} - \frac{171 x ^{4} 4 \frac{x}{1220703125}}{2 5 ^{23} \cdot 625} + \frac{19 x ^{9}}{5 ^{51} 1220703125} - \frac{x ^{4} ( 4 \frac{x}{1220703125} ) ^{3}}{122070312 5 ^{4}}

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