développer (5e^3+3f^2)^6

Solution

développer

(5 e^{3} + 3 f^{2})^{6}

729 f^{12} + 7290 e^{3} f^{10} + 30375 e^{6} f^{8} + 67500 e^{9} f^{6} + 84375 e^{12} f^{4} + 56250 e^{15} f^{2} + 15625 e^{18}

étapes des solutions

(5 e^{3} + 3 f^{2})^{6}

Appliquer le théorème du binôme:

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

a = 5 e^{3}, b = 3 f^{2}

= i = 0 \sum 6 (i 6) (5 e^{3})^{(6 - i)} (3 f^{2})^{i}

Développer la somme

= \frac{6 !}{0 ! ( 6 - 0 ) !} (5 e^{3})^{6} (3 f^{2})^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (5 e^{3})^{5} (3 f^{2})^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (5 e^{3})^{4} (3 f^{2})^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (5 e^{3})^{3} (3 f^{2})^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (5 e^{3})^{2} (3 f^{2})^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (5 e^{3})^{1} (3 f^{2})^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (5 e^{3})^{0} (3 f^{2})^{6}

\frac{6 !}{0 ! ( 6 - 0 ) !} (5 e^{3})^{6} (3 f^{2})^{0} = 15625 e^{18}

Simplifier

\frac{6 !}{1 ! ( 6 - 1 ) !} (5 e^{3})^{5} (3 f^{2})^{1} : 56250 e^{15} f^{2}

Simplifier

\frac{6 !}{2 ! ( 6 - 2 ) !} (5 e^{3})^{4} (3 f^{2})^{2} : 84375 e^{12} f^{4}

Simplifier

\frac{6 !}{3 ! ( 6 - 3 ) !} (5 e^{3})^{3} (3 f^{2})^{3} : 67500 e^{9} f^{6}

Simplifier

\frac{6 !}{4 ! ( 6 - 4 ) !} (5 e^{3})^{2} (3 f^{2})^{4} : 30375 e^{6} f^{8}

Simplifier

\frac{6 !}{5 ! ( 6 - 5 ) !} (5 e^{3})^{1} (3 f^{2})^{5} : 7290 e^{3} f^{10}

Simplifier

\frac{6 !}{6 ! ( 6 - 6 ) !} (5 e^{3})^{0} (3 f^{2})^{6} : 729 f^{12}

= 15625 e^{18} + 56250 e^{15} f^{2} + 84375 e^{12} f^{4} + 67500 e^{9} f^{6} + 30375 e^{6} f^{8} + 7290 e^{3} f^{10} + 729 f^{12}

Récrire sous la forme standard

= 729 f^{12} + 7290 e^{3} f^{10} + 30375 e^{6} f^{8} + 67500 e^{9} f^{6} + 84375 e^{12} f^{4} + 56250 e^{15} f^{2} + 15625 e^{18}

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