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Popular Álgebra >

expandir (n^3+2m^3)^6

Solução

expandir

(n^{3} + 2 m^{3})^{6}

Solução

n^{18} + 12 n^{15} m^{3} + 60 n^{12} m^{6} + 160 n^{9} m^{9} + 240 n^{6} m^{12} + 192 n^{3} m^{15} + 64 m^{18}

Passos da solução

(n^{3} + 2 m^{3})^{6}

Aplicar o teorema do binômio:

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

a = n^{3}, b = 2 m^{3}

= i = 0 \sum 6 (i 6) (n^{3})^{(6 - i)} (2 m^{3})^{i}

Expandir a soma

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

Simplificar

\frac{6 !}{0 ! ( 6 - 0 ) !} (n^{3})^{6} (2 m^{3})^{0} : n^{18}

Simplificar

\frac{6 !}{1 ! ( 6 - 1 ) !} (n^{3})^{5} (2 m^{3})^{1} : 12 n^{15} m^{3}

Simplificar

\frac{6 !}{2 ! ( 6 - 2 ) !} (n^{3})^{4} (2 m^{3})^{2} : 60 n^{12} m^{6}

Simplificar

\frac{6 !}{3 ! ( 6 - 3 ) !} (n^{3})^{3} (2 m^{3})^{3} : 160 n^{9} m^{9}

Simplificar

\frac{6 !}{4 ! ( 6 - 4 ) !} (n^{3})^{2} (2 m^{3})^{4} : 240 n^{6} m^{12}

Simplificar

\frac{6 !}{5 ! ( 6 - 5 ) !} (n^{3})^{1} (2 m^{3})^{5} : 192 n^{3} m^{15}

Simplificar

\frac{6 !}{6 ! ( 6 - 6 ) !} (n^{3})^{0} (2 m^{3})^{6} : 64 m^{18}

= n^{18} + 12 n^{15} m^{3} + 60 n^{12} m^{6} + 160 n^{9} m^{9} + 240 n^{6} m^{12} + 192 n^{3} m^{15} + 64 m^{18}

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