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Popular Álgebra >

expandir (5x^2-11y^7)^{25}

Solução

expandir

(5 x^{2} - 11 y^{7})^{25}

Solução

5^{25} x^{50} - 5^{24} \cdot 275 x^{48} y^{7} + 5^{23} \cdot 36300 x^{46} y^{14} - 5^{22} \cdot 3061300 x^{44} y^{21} + 5^{21} \cdot 185208650 x^{42} y^{28} - 5^{20} \cdot 8556639630 x^{40} y^{35} + 5^{19} \cdot 313743453100 x^{38} y^{42} - 5^{18} \cdot 9367483099700 x^{36} y^{49} + \frac{5 ^{16} \cdot 9.348 E 18 x ^{34} y ^{56}}{8064} - 5^{16} \cdot 1 1^{9} \cdot 2042975 x^{32} y^{63} + 5^{15} \cdot 1 1^{10} \cdot 3268760 x^{30} y^{70} - 5^{14} \cdot 1 1^{11} \cdot 4457400 x^{28} y^{77} + 5^{13} \cdot 1 1^{12} \cdot 5200300 x^{26} y^{84} - \frac{1 1 ^{12} \cdot 6.08143 E 23 x ^{24} y ^{91}}{43545600} + \frac{1 1 ^{13} \cdot 8.68775 E 21 x ^{22} y ^{98} - 1 1 ^{15} \cdot 1.15837 E 20 x ^{20} y ^{105}}{3628800} + \frac{1 1 ^{16} \cdot 1.13122 E 16 x ^{18} y ^{112}}{2835} - 1 1^{17} \cdot 422490234375 x^{16} y^{119} + 1 1^{18} \cdot 37554687500 x^{14} y^{126} - 1 1^{19} \cdot 2767187500 x^{12} y^{133} + 1 1^{20} \cdot 166031250 x^{10} y^{140} - 1 1^{21} \cdot 7906250 x^{8} y^{147} + 1 1^{22} \cdot 287500 x^{6} y^{154} - 1 1^{23} \cdot 7500 x^{4} y^{161} + 1 1^{24} \cdot 125 x^{2} y^{168} - 1 1^{25} y^{175}

Passos da solução

(5 x^{2} - 11 y^{7})^{25}

Aplicar o teorema do binômio:

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

a = 5 x^{2}, b = - 11 y^{7}

= i = 0 \sum 25 (i 25) (5 x^{2})^{(25 - i)} (- 11 y^{7})^{i}

Expandir a soma

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

Simplificar

\frac{25 !}{0 ! ( 25 - 0 ) !} (5 x^{2})^{25} (- 11 y^{7})^{0} : 5^{25} x^{50}

Simplificar

\frac{25 !}{1 ! ( 25 - 1 ) !} (5 x^{2})^{24} (- 11 y^{7})^{1} : - 5^{24} \cdot 275 x^{48} y^{7}

Simplificar

\frac{25 !}{2 ! ( 25 - 2 ) !} (5 x^{2})^{23} (- 11 y^{7})^{2} : 5^{23} \cdot 36300 x^{46} y^{14}

Simplificar

\frac{25 !}{3 ! ( 25 - 3 ) !} (5 x^{2})^{22} (- 11 y^{7})^{3} : - 5^{22} \cdot 3061300 x^{44} y^{21}

Simplificar

\frac{25 !}{4 ! ( 25 - 4 ) !} (5 x^{2})^{21} (- 11 y^{7})^{4} : 5^{21} \cdot 185208650 x^{42} y^{28}

Simplificar

\frac{25 !}{5 ! ( 25 - 5 ) !} (5 x^{2})^{20} (- 11 y^{7})^{5} : - 5^{20} \cdot 8556639630 x^{40} y^{35}

Simplificar

\frac{25 !}{6 ! ( 25 - 6 ) !} (5 x^{2})^{19} (- 11 y^{7})^{6} : 5^{19} \cdot 313743453100 x^{38} y^{42}

Simplificar

\frac{25 !}{7 ! ( 25 - 7 ) !} (5 x^{2})^{18} (- 11 y^{7})^{7} : - 5^{18} \cdot 9367483099700 x^{36} y^{49}

Simplificar

\frac{25 !}{8 ! ( 25 - 8 ) !} (5 x^{2})^{17} (- 11 y^{7})^{8} : \frac{5 ^{16} \cdot 9.348 E 18 x ^{34} y ^{56}}{8064}

Simplificar

\frac{25 !}{9 ! ( 25 - 9 ) !} (5 x^{2})^{16} (- 11 y^{7})^{9} : - 5^{16} \cdot 1 1^{9} \cdot 2042975 x^{32} y^{63}

Simplificar

\frac{25 !}{10 ! ( 25 - 10 ) !} (5 x^{2})^{15} (- 11 y^{7})^{10} : 5^{15} \cdot 1 1^{10} \cdot 3268760 x^{30} y^{70}

Simplificar

\frac{25 !}{11 ! ( 25 - 11 ) !} (5 x^{2})^{14} (- 11 y^{7})^{11} : - 5^{14} \cdot 1 1^{11} \cdot 4457400 x^{28} y^{77}

Simplificar

\frac{25 !}{12 ! ( 25 - 12 ) !} (5 x^{2})^{13} (- 11 y^{7})^{12} : 5^{13} \cdot 1 1^{12} \cdot 5200300 x^{26} y^{84}

Simplificar

\frac{25 !}{13 ! ( 25 - 13 ) !} (5 x^{2})^{12} (- 11 y^{7})^{13} : - \frac{1 1 ^{13} \cdot 6.08143 E 23 x ^{24} y ^{91}}{479001600}

Simplificar

\frac{25 !}{14 ! ( 25 - 14 ) !} (5 x^{2})^{11} (- 11 y^{7})^{14} : \frac{1 1 ^{13} \cdot 8.68775 E 21 x ^{22} y ^{98}}{3628800}

Simplificar

\frac{25 !}{15 ! ( 25 - 15 ) !} (5 x^{2})^{10} (- 11 y^{7})^{15} : - \frac{1 1 ^{15} \cdot 1.15837 E 20 x ^{20} y ^{105}}{3628800}

Simplificar

\frac{25 !}{16 ! ( 25 - 16 ) !} (5 x^{2})^{9} (- 11 y^{7})^{16} : \frac{1 1 ^{16} \cdot 1.13122 E 16 x ^{18} y ^{112}}{2835}

Simplificar

\frac{25 !}{17 ! ( 25 - 17 ) !} (5 x^{2})^{8} (- 11 y^{7})^{17} : - 1 1^{17} \cdot 422490234375 x^{16} y^{119}

Simplificar

\frac{25 !}{18 ! ( 25 - 18 ) !} (5 x^{2})^{7} (- 11 y^{7})^{18} : 1 1^{18} \cdot 37554687500 x^{14} y^{126}

Simplificar

\frac{25 !}{19 ! ( 25 - 19 ) !} (5 x^{2})^{6} (- 11 y^{7})^{19} : - 1 1^{19} \cdot 2767187500 x^{12} y^{133}

Simplificar

\frac{25 !}{20 ! ( 25 - 20 ) !} (5 x^{2})^{5} (- 11 y^{7})^{20} : 1 1^{20} \cdot 166031250 x^{10} y^{140}

Simplificar

\frac{25 !}{21 ! ( 25 - 21 ) !} (5 x^{2})^{4} (- 11 y^{7})^{21} : - 1 1^{21} \cdot 7906250 x^{8} y^{147}

Simplificar

\frac{25 !}{22 ! ( 25 - 22 ) !} (5 x^{2})^{3} (- 11 y^{7})^{22} : 1 1^{22} \cdot 287500 x^{6} y^{154}

Simplificar

\frac{25 !}{23 ! ( 25 - 23 ) !} (5 x^{2})^{2} (- 11 y^{7})^{23} : - 1 1^{23} \cdot 7500 x^{4} y^{161}

Simplificar

\frac{25 !}{24 ! ( 25 - 24 ) !} (5 x^{2})^{1} (- 11 y^{7})^{24} : 1 1^{24} \cdot 125 x^{2} y^{168}

Simplificar

\frac{25 !}{25 ! ( 25 - 25 ) !} (5 x^{2})^{0} (- 11 y^{7})^{25} : - 1 1^{25} y^{175}

= 5^{25} x^{50} - 5^{24} \cdot 275 x^{48} y^{7} + 5^{23} \cdot 36300 x^{46} y^{14} - 5^{22} \cdot 3061300 x^{44} y^{21} + 5^{21} \cdot 185208650 x^{42} y^{28} - 5^{20} \cdot 8556639630 x^{40} y^{35} + 5^{19} \cdot 313743453100 x^{38} y^{42} - 5^{18} \cdot 9367483099700 x^{36} y^{49} + \frac{5 ^{16} \cdot 9.348 E 18 x ^{34} y ^{56}}{8064} - 5^{16} \cdot 1 1^{9} \cdot 2042975 x^{32} y^{63} + 5^{15} \cdot 1 1^{10} \cdot 3268760 x^{30} y^{70} - 5^{14} \cdot 1 1^{11} \cdot 4457400 x^{28} y^{77} + 5^{13} \cdot 1 1^{12} \cdot 5200300 x^{26} y^{84} - \frac{1 1 ^{13} \cdot 6.08143 E 23 x ^{24} y ^{91}}{479001600} + \frac{1 1 ^{13} \cdot 8.68775 E 21 x ^{22} y ^{98}}{3628800} - \frac{1 1 ^{15} \cdot 1.15837 E 20 x ^{20} y ^{105}}{3628800} + \frac{1 1 ^{16} \cdot 1.13122 E 16 x ^{18} y ^{112}}{2835} - 1 1^{17} \cdot 422490234375 x^{16} y^{119} + 1 1^{18} \cdot 37554687500 x^{14} y^{126} - 1 1^{19} \cdot 2767187500 x^{12} y^{133} + 1 1^{20} \cdot 166031250 x^{10} y^{140} - 1 1^{21} \cdot 7906250 x^{8} y^{147} + 1 1^{22} \cdot 287500 x^{6} y^{154} - 1 1^{23} \cdot 7500 x^{4} y^{161} + 1 1^{24} \cdot 125 x^{2} y^{168} - 1 1^{25} y^{175}

Fatorar

= 5^{25} x^{50} - 5^{24} \cdot 275 x^{48} y^{7} + 5^{23} \cdot 36300 x^{46} y^{14} - 5^{22} \cdot 3061300 x^{44} y^{21} + 5^{21} \cdot 185208650 x^{42} y^{28} - 5^{20} \cdot 8556639630 x^{40} y^{35} + 5^{19} \cdot 313743453100 x^{38} y^{42} - 5^{18} \cdot 9367483099700 x^{36} y^{49} + \frac{5 ^{16} \cdot 9.348 E 18 x ^{34} y ^{56}}{8064} - 5^{16} \cdot 1 1^{9} \cdot 2042975 x^{32} y^{63} + 5^{15} \cdot 1 1^{10} \cdot 3268760 x^{30} y^{70} - 5^{14} \cdot 1 1^{11} \cdot 4457400 x^{28} y^{77} + 5^{13} \cdot 1 1^{12} \cdot 5200300 x^{26} y^{84} - \frac{1 1 ^{12} \cdot 6.08143 E 23 x ^{24} y ^{91}}{43545600} + \frac{1 1 ^{13} \cdot 8.68775 E 21 x ^{22} y ^{98} - 1 1 ^{15} \cdot 1.15837 E 20 x ^{20} y ^{105}}{3628800} + \frac{1 1 ^{16} \cdot 1.13122 E 16 x ^{18} y ^{112}}{2835} - 1 1^{17} \cdot 422490234375 x^{16} y^{119} + 1 1^{18} \cdot 37554687500 x^{14} y^{126} - 1 1^{19} \cdot 2767187500 x^{12} y^{133} + 1 1^{20} \cdot 166031250 x^{10} y^{140} - 1 1^{21} \cdot 7906250 x^{8} y^{147} + 1 1^{22} \cdot 287500 x^{6} y^{154} - 1 1^{23} \cdot 7500 x^{4} y^{161} + 1 1^{24} \cdot 125 x^{2} y^{168} - 1 1^{25} y^{175}

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