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Popolare Algebra >

espandere (5e^3+3p^2)^6

Soluzione

espandere

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

Soluzione

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

Fasi della soluzione

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

Applicare il teorema binomiale:

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

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

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

Espandere la sommatoria

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

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

Semplifica

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

Semplifica

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

Semplifica

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

Semplifica

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

Semplifica

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

Semplifica

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

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

Riscrivere in forma standard

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

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