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Popular Álgebra >

expandir (2x+3y)^6

Solução

expandir

(2 x + 3 y)^{6}

Solução

64 x^{6} + 576 x^{5} y + 2160 x^{4} y^{2} + 4320 x^{3} y^{3} + 4860 x^{2} y^{4} + 2916 x y^{5} + 729 y^{6}

Passos da solução

(2 x + 3 y)^{6}

Aplicar o teorema do binômio:

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

a = 2 x, b = 3 y

= i = 0 \sum 6 (i 6) (2 x)^{(6 - i)} (3 y)^{i}

Expandir a soma

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

Simplificar

\frac{6 !}{0 ! ( 6 - 0 ) !} (2 x)^{6} (3 y)^{0} : 64 x^{6}

Simplificar

\frac{6 !}{1 ! ( 6 - 1 ) !} (2 x)^{5} (3 y)^{1} : 576 x^{5} y

Simplificar

\frac{6 !}{2 ! ( 6 - 2 ) !} (2 x)^{4} (3 y)^{2} : 2160 x^{4} y^{2}

Simplificar

\frac{6 !}{3 ! ( 6 - 3 ) !} (2 x)^{3} (3 y)^{3} : 4320 x^{3} y^{3}

Simplificar

\frac{6 !}{4 ! ( 6 - 4 ) !} (2 x)^{2} (3 y)^{4} : 4860 x^{2} y^{4}

Simplificar

\frac{6 !}{5 ! ( 6 - 5 ) !} (2 x)^{1} (3 y)^{5} : 2916 x y^{5}

Simplificar

\frac{6 !}{6 ! ( 6 - 6 ) !} (2 x)^{0} (3 y)^{6} : 729 y^{6}

= 64 x^{6} + 576 x^{5} y + 2160 x^{4} y^{2} + 4320 x^{3} y^{3} + 4860 x^{2} y^{4} + 2916 x y^{5} + 729 y^{6}

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