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desarrollar (5-\sqrt[5]{x/(244140625)})^{19}

Solución

desarrollar

(5 - 5 \frac{x}{244140625})^{19}

Solución

5^{19} - 5 5^{78} \cdot 19 5 x + 5^{17} \cdot 171 (5 \frac{x}{244140625})^{2} - 5^{16} \cdot 969 (5 \frac{x}{244140625})^{3} + 5^{15} \cdot 3876 (5 \frac{x}{244140625})^{4} - 290700 x + 135660 x 5 \frac{x}{244140625} - \frac{50388 x 5 x ^{2}}{5 24414062 5 ^{2}} + \frac{75582 x 5 x ^{3}}{5 5 24414062 5 ^{3}} - \frac{92378 x 5 x ^{4}}{25 5 24414062 5 ^{4}} + \frac{92378 x ^{2}}{30517578125} - \frac{75582 5 x ^{11}}{390625 \cdot 62 5 ^{2} 5 244140625} + \frac{50388 x ^{2} 5 x ^{2}}{78125 \cdot 312 5 ^{2} 5 24414062 5 ^{2}} - \frac{27132 x ^{2} 5 x ^{3}}{1562 5 ^{3} 5 24414062 5 ^{3}} + \frac{11628 x ^{2} 5 x ^{4}}{312 5 ^{3} \cdot 625 5 24414062 5 ^{4}} - \frac{3876 x ^{3}}{62 5 ^{8}} + \frac{969 5 x ^{16}}{12 5 ^{11} 5 244140625} - \frac{171 x ^{3} 5 x ^{2}}{2 5 ^{17} 5 24414062 5 ^{2}} + \frac{19 x ^{3} 5 x ^{3}}{5 ^{35} 5 24414062 5 ^{3}} - \frac{x ^{3} ( 5 \frac{x}{244140625} ) ^{4}}{24414062 5 ^{3}}

Pasos de solución

(5 - 5 \frac{x}{244140625})^{19}

Aplicar el teorema del binomio:

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

a = 5, b = - 5 \frac{x}{244140625}

= i = 0 \sum 19 (i 19) \cdot 5^{(19 - i)} (- 5 \frac{x}{244140625})^{i}

Expandir sumatorio

= \frac{19 !}{0 ! ( 19 - 0 ) !} \cdot 5^{19} (- 5 \frac{x}{244140625})^{0} + \frac{19 !}{1 ! ( 19 - 1 ) !} \cdot 5^{18} (- 5 \frac{x}{244140625})^{1} + \frac{19 !}{2 ! ( 19 - 2 ) !} \cdot 5^{17} (- 5 \frac{x}{244140625})^{2} + \frac{19 !}{3 ! ( 19 - 3 ) !} \cdot 5^{16} (- 5 \frac{x}{244140625})^{3} + \frac{19 !}{4 ! ( 19 - 4 ) !} \cdot 5^{15} (- 5 \frac{x}{244140625})^{4} + \frac{19 !}{5 ! ( 19 - 5 ) !} \cdot 5^{14} (- 5 \frac{x}{244140625})^{5} + \frac{19 !}{6 ! ( 19 - 6 ) !} \cdot 5^{13} (- 5 \frac{x}{244140625})^{6} + \frac{19 !}{7 ! ( 19 - 7 ) !} \cdot 5^{12} (- 5 \frac{x}{244140625})^{7} + \frac{19 !}{8 ! ( 19 - 8 ) !} \cdot 5^{11} (- 5 \frac{x}{244140625})^{8} + \frac{19 !}{9 ! ( 19 - 9 ) !} \cdot 5^{10} (- 5 \frac{x}{244140625})^{9} + \frac{19 !}{10 ! ( 19 - 10 ) !} \cdot 5^{9} (- 5 \frac{x}{244140625})^{10} + \frac{19 !}{11 ! ( 19 - 11 ) !} \cdot 5^{8} (- 5 \frac{x}{244140625})^{11} + \frac{19 !}{12 ! ( 19 - 12 ) !} \cdot 5^{7} (- 5 \frac{x}{244140625})^{12} + \frac{19 !}{13 ! ( 19 - 13 ) !} \cdot 5^{6} (- 5 \frac{x}{244140625})^{13} + \frac{19 !}{14 ! ( 19 - 14 ) !} \cdot 5^{5} (- 5 \frac{x}{244140625})^{14} + \frac{19 !}{15 ! ( 19 - 15 ) !} \cdot 5^{4} (- 5 \frac{x}{244140625})^{15} + \frac{19 !}{16 ! ( 19 - 16 ) !} \cdot 5^{3} (- 5 \frac{x}{244140625})^{16} + \frac{19 !}{17 ! ( 19 - 17 ) !} \cdot 5^{2} (- 5 \frac{x}{244140625})^{17} + \frac{19 !}{18 ! ( 19 - 18 ) !} \cdot 5^{1} (- 5 \frac{x}{244140625})^{18} + \frac{19 !}{19 ! ( 19 - 19 ) !} \cdot 5^{0} (- 5 \frac{x}{244140625})^{19}

Simplificar

\frac{19 !}{0 ! ( 19 - 0 ) !} \cdot 5^{19} (- 5 \frac{x}{244140625})^{0} : 5^{19}

Simplificar

\frac{19 !}{1 ! ( 19 - 1 ) !} \cdot 5^{18} (- 5 \frac{x}{244140625})^{1} : - \frac{5 ^{18} \cdot 19 5 x}{5 244140625}

Simplificar

\frac{19 !}{2 ! ( 19 - 2 ) !} \cdot 5^{17} (- 5 \frac{x}{244140625})^{2} : 5^{17} \cdot 171 (5 \frac{x}{244140625})^{2}

Simplificar

\frac{19 !}{3 ! ( 19 - 3 ) !} \cdot 5^{16} (- 5 \frac{x}{244140625})^{3} : - 5^{16} \cdot 969 (5 \frac{x}{244140625})^{3}

Simplificar

\frac{19 !}{4 ! ( 19 - 4 ) !} \cdot 5^{15} (- 5 \frac{x}{244140625})^{4} : 5^{15} \cdot 3876 (5 \frac{x}{244140625})^{4}

Simplificar

\frac{19 !}{5 ! ( 19 - 5 ) !} \cdot 5^{14} (- 5 \frac{x}{244140625})^{5} : - 290700 x

Simplificar

\frac{19 !}{6 ! ( 19 - 6 ) !} \cdot 5^{13} (- 5 \frac{x}{244140625})^{6} : 5^{13} \cdot 27132 (5 \frac{x}{244140625})^{6}

Simplificar

\frac{19 !}{7 ! ( 19 - 7 ) !} \cdot 5^{12} (- 5 \frac{x}{244140625})^{7} : - 1.23018 E 13 (5 \frac{x}{244140625})^{7}

Simplificar

\frac{19 !}{8 ! ( 19 - 8 ) !} \cdot 5^{11} (- 5 \frac{x}{244140625})^{8} : 3690527343750 (5 \frac{x}{244140625})^{8}

Simplificar

\frac{19 !}{9 ! ( 19 - 9 ) !} \cdot 5^{10} (- 5 \frac{x}{244140625})^{9} : - 902128906250 (5 \frac{x}{244140625})^{9}

Simplificar

\frac{19 !}{10 ! ( 19 - 10 ) !} \cdot 5^{9} (- 5 \frac{x}{244140625})^{10} : \frac{180425781250 x ^{2}}{24414062 5 ^{2}}

Simplificar

\frac{19 !}{11 ! ( 19 - 11 ) !} \cdot 5^{8} (- 5 \frac{x}{244140625})^{11} : - 29524218750 (5 \frac{x}{244140625})^{11}

Simplificar

\frac{19 !}{12 ! ( 19 - 12 ) !} \cdot 5^{7} (- 5 \frac{x}{244140625})^{12} : 3936562500 (5 \frac{x}{244140625})^{12}

Simplificar

\frac{19 !}{13 ! ( 19 - 13 ) !} \cdot 5^{6} (- 5 \frac{x}{244140625})^{13} : - 423937500 (5 \frac{x}{244140625})^{13}

Simplificar

\frac{19 !}{14 ! ( 19 - 14 ) !} \cdot 5^{5} (- 5 \frac{x}{244140625})^{14} : 36337500 (5 \frac{x}{244140625})^{14}

Simplificar

\frac{19 !}{15 ! ( 19 - 15 ) !} \cdot 5^{4} (- 5 \frac{x}{244140625})^{15} : - \frac{2422500 x ^{3}}{24414062 5 ^{3}}

Simplificar

\frac{19 !}{16 ! ( 19 - 16 ) !} \cdot 5^{3} (- 5 \frac{x}{244140625})^{16} : 121125 (5 \frac{x}{244140625})^{16}

Simplificar

\frac{19 !}{17 ! ( 19 - 17 ) !} \cdot 5^{2} (- 5 \frac{x}{244140625})^{17} : - 4275 (5 \frac{x}{244140625})^{17}

Simplificar

\frac{19 !}{18 ! ( 19 - 18 ) !} \cdot 5^{1} (- 5 \frac{x}{244140625})^{18} : 95 (5 \frac{x}{244140625})^{18}

Simplificar

\frac{19 !}{19 ! ( 19 - 19 ) !} \cdot 5^{0} (- 5 \frac{x}{244140625})^{19} : - (5 \frac{x}{244140625})^{19}

= 5^{19} - \frac{5 ^{18} \cdot 19 5 x}{5 244140625} + 5^{17} \cdot 171 (5 \frac{x}{244140625})^{2} - 5^{16} \cdot 969 (5 \frac{x}{244140625})^{3} + 5^{15} \cdot 3876 (5 \frac{x}{244140625})^{4} - 290700 x + 5^{13} \cdot 27132 (5 \frac{x}{244140625})^{6} - 1.23018 E 13 (5 \frac{x}{244140625})^{7} + 3690527343750 (5 \frac{x}{244140625})^{8} - 902128906250 (5 \frac{x}{244140625})^{9} + \frac{180425781250 x ^{2}}{24414062 5 ^{2}} - 29524218750 (5 \frac{x}{244140625})^{11} + 3936562500 (5 \frac{x}{244140625})^{12} - 423937500 (5 \frac{x}{244140625})^{13} + 36337500 (5 \frac{x}{244140625})^{14} - \frac{2422500 x ^{3}}{24414062 5 ^{3}} + 121125 (5 \frac{x}{244140625})^{16} - 4275 (5 \frac{x}{244140625})^{17} + 95 (5 \frac{x}{244140625})^{18} - (5 \frac{x}{244140625})^{19}

Simplificar

5^{19} - \frac{5 ^{18} \cdot 19 5 x}{5 244140625} + 5^{17} \cdot 171 (5 \frac{x}{244140625})^{2} - 5^{16} \cdot 969 (5 \frac{x}{244140625})^{3} + 5^{15} \cdot 3876 (5 \frac{x}{244140625})^{4} - 290700 x + 5^{13} \cdot 27132 (5 \frac{x}{244140625})^{6} - 1.23018 E 13 (5 \frac{x}{244140625})^{7} + 3690527343750 (5 \frac{x}{244140625})^{8} - 902128906250 (5 \frac{x}{244140625})^{9} + \frac{180425781250 x ^{2}}{24414062 5 ^{2}} - 29524218750 (5 \frac{x}{244140625})^{11} + 3936562500 (5 \frac{x}{244140625})^{12} - 423937500 (5 \frac{x}{244140625})^{13} + 36337500 (5 \frac{x}{244140625})^{14} - \frac{2422500 x ^{3}}{24414062 5 ^{3}} + 121125 (5 \frac{x}{244140625})^{16} - 4275 (5 \frac{x}{244140625})^{17} + 95 (5 \frac{x}{244140625})^{18} - (5 \frac{x}{244140625})^{19} : 5^{19} - 5^{\frac{78}{5}} \cdot 19 5 x + 5^{17} \cdot 171 (5 \frac{x}{244140625})^{2} - 5^{16} \cdot 969 (5 \frac{x}{244140625})^{3} + 5^{15} \cdot 3876 (5 \frac{x}{244140625})^{4} - 290700 x + 135660 x 5 \frac{x}{244140625} - \frac{50388 x 5 x ^{2}}{5 24414062 5 ^{2}} + \frac{75582 x 5 x ^{3}}{5 5 24414062 5 ^{3}} - \frac{92378 x 5 x ^{4}}{25 5 24414062 5 ^{4}} + \frac{92378 x ^{2}}{30517578125} - \frac{75582 x ^{\frac{11}{5}}}{390625 \cdot 62 5 ^{2} 5 244140625} + \frac{50388 x ^{2} 5 x ^{2}}{78125 \cdot 312 5 ^{2} 5 24414062 5 ^{2}} - \frac{27132 x ^{2} 5 x ^{3}}{1562 5 ^{3} 5 24414062 5 ^{3}} + \frac{11628 x ^{2} 5 x ^{4}}{312 5 ^{3} \cdot 625 5 24414062 5 ^{4}} - \frac{3876 x ^{3}}{62 5 ^{8}} + \frac{969 x ^{\frac{16}{5}}}{12 5 ^{11} 5 244140625} - \frac{171 x ^{3} 5 x ^{2}}{2 5 ^{17} 5 24414062 5 ^{2}} + \frac{19 x ^{3} 5 x ^{3}}{5 ^{35} 5 24414062 5 ^{3}} - \frac{x ^{3} ( 5 \frac{x}{244140625} ) ^{4}}{24414062 5 ^{3}}

= 5^{19} - 5^{\frac{78}{5}} \cdot 19 5 x + 5^{17} \cdot 171 (5 \frac{x}{244140625})^{2} - 5^{16} \cdot 969 (5 \frac{x}{244140625})^{3} + 5^{15} \cdot 3876 (5 \frac{x}{244140625})^{4} - 290700 x + 135660 x 5 \frac{x}{244140625} - \frac{50388 x 5 x ^{2}}{5 24414062 5 ^{2}} + \frac{75582 x 5 x ^{3}}{5 5 24414062 5 ^{3}} - \frac{92378 x 5 x ^{4}}{25 5 24414062 5 ^{4}} + \frac{92378 x ^{2}}{30517578125} - \frac{75582 x ^{\frac{11}{5}}}{390625 \cdot 62 5 ^{2} 5 244140625} + \frac{50388 x ^{2} 5 x ^{2}}{78125 \cdot 312 5 ^{2} 5 24414062 5 ^{2}} - \frac{27132 x ^{2} 5 x ^{3}}{1562 5 ^{3} 5 24414062 5 ^{3}} + \frac{11628 x ^{2} 5 x ^{4}}{312 5 ^{3} \cdot 625 5 24414062 5 ^{4}} - \frac{3876 x ^{3}}{62 5 ^{8}} + \frac{969 x ^{\frac{16}{5}}}{12 5 ^{11} 5 244140625} - \frac{171 x ^{3} 5 x ^{2}}{2 5 ^{17} 5 24414062 5 ^{2}} + \frac{19 x ^{3} 5 x ^{3}}{5 ^{35} 5 24414062 5 ^{3}} - \frac{x ^{3} ( 5 \frac{x}{244140625} ) ^{4}}{24414062 5 ^{3}}

Simplificar

5^{19} - 5^{\frac{78}{5}} \cdot 19 5 x + 5^{17} \cdot 171 (5 \frac{x}{244140625})^{2} - 5^{16} \cdot 969 (5 \frac{x}{244140625})^{3} + 5^{15} \cdot 3876 (5 \frac{x}{244140625})^{4} - 290700 x + 135660 x 5 \frac{x}{244140625} - \frac{50388 x 5 x ^{2}}{5 24414062 5 ^{2}} + \frac{75582 x 5 x ^{3}}{5 5 24414062 5 ^{3}} - \frac{92378 x 5 x ^{4}}{25 5 24414062 5 ^{4}} + \frac{92378 x ^{2}}{30517578125} - \frac{75582 x ^{\frac{11}{5}}}{390625 \cdot 62 5 ^{2} 5 244140625} + \frac{50388 x ^{2} 5 x ^{2}}{78125 \cdot 312 5 ^{2} 5 24414062 5 ^{2}} - \frac{27132 x ^{2} 5 x ^{3}}{1562 5 ^{3} 5 24414062 5 ^{3}} + \frac{11628 x ^{2} 5 x ^{4}}{312 5 ^{3} \cdot 625 5 24414062 5 ^{4}} - \frac{3876 x ^{3}}{62 5 ^{8}} + \frac{969 x ^{\frac{16}{5}}}{12 5 ^{11} 5 244140625} - \frac{171 x ^{3} 5 x ^{2}}{2 5 ^{17} 5 24414062 5 ^{2}} + \frac{19 x ^{3} 5 x ^{3}}{5 ^{35} 5 24414062 5 ^{3}} - \frac{x ^{3} ( 5 \frac{x}{244140625} ) ^{4}}{24414062 5 ^{3}} : 5^{19} - 5 5^{78} \cdot 19 5 x + 5^{17} \cdot 171 (5 \frac{x}{244140625})^{2} - 5^{16} \cdot 969 (5 \frac{x}{244140625})^{3} + 5^{15} \cdot 3876 (5 \frac{x}{244140625})^{4} - 290700 x + 135660 x 5 \frac{x}{244140625} - \frac{50388 x 5 x ^{2}}{5 24414062 5 ^{2}} + \frac{75582 x 5 x ^{3}}{5 5 24414062 5 ^{3}} - \frac{92378 x 5 x ^{4}}{25 5 24414062 5 ^{4}} + \frac{92378 x ^{2}}{30517578125} - \frac{75582 5 x ^{11}}{390625 \cdot 62 5 ^{2} 5 244140625} + \frac{50388 x ^{2} 5 x ^{2}}{78125 \cdot 312 5 ^{2} 5 24414062 5 ^{2}} - \frac{27132 x ^{2} 5 x ^{3}}{1562 5 ^{3} 5 24414062 5 ^{3}} + \frac{11628 x ^{2} 5 x ^{4}}{312 5 ^{3} \cdot 625 5 24414062 5 ^{4}} - \frac{3876 x ^{3}}{62 5 ^{8}} + \frac{969 5 x ^{16}}{12 5 ^{11} 5 244140625} - \frac{171 x ^{3} 5 x ^{2}}{2 5 ^{17} 5 24414062 5 ^{2}} + \frac{19 x ^{3} 5 x ^{3}}{5 ^{35} 5 24414062 5 ^{3}} - \frac{x ^{3} ( 5 \frac{x}{244140625} ) ^{4}}{24414062 5 ^{3}}

= 5^{19} - 5 5^{78} \cdot 19 5 x + 5^{17} \cdot 171 (5 \frac{x}{244140625})^{2} - 5^{16} \cdot 969 (5 \frac{x}{244140625})^{3} + 5^{15} \cdot 3876 (5 \frac{x}{244140625})^{4} - 290700 x + 135660 x 5 \frac{x}{244140625} - \frac{50388 x 5 x ^{2}}{5 24414062 5 ^{2}} + \frac{75582 x 5 x ^{3}}{5 5 24414062 5 ^{3}} - \frac{92378 x 5 x ^{4}}{25 5 24414062 5 ^{4}} + \frac{92378 x ^{2}}{30517578125} - \frac{75582 5 x ^{11}}{390625 \cdot 62 5 ^{2} 5 244140625} + \frac{50388 x ^{2} 5 x ^{2}}{78125 \cdot 312 5 ^{2} 5 24414062 5 ^{2}} - \frac{27132 x ^{2} 5 x ^{3}}{1562 5 ^{3} 5 24414062 5 ^{3}} + \frac{11628 x ^{2} 5 x ^{4}}{312 5 ^{3} \cdot 625 5 24414062 5 ^{4}} - \frac{3876 x ^{3}}{62 5 ^{8}} + \frac{969 5 x ^{16}}{12 5 ^{11} 5 244140625} - \frac{171 x ^{3} 5 x ^{2}}{2 5 ^{17} 5 24414062 5 ^{2}} + \frac{19 x ^{3} 5 x ^{3}}{5 ^{35} 5 24414062 5 ^{3}} - \frac{x ^{3} ( 5 \frac{x}{244140625} ) ^{4}}{24414062 5 ^{3}}

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