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Popular Álgebra >

expandir (5x+3y^2)^7

Solução

expandir

(5 x + 3 y^{2})^{7}

Solução

78125 x^{7} + 328125 x^{6} y^{2} + 590625 x^{5} y^{4} + 590625 x^{4} y^{6} + 354375 x^{3} y^{8} + 127575 x^{2} y^{10} + 25515 x y^{12} + 2187 y^{14}

Passos da solução

(5 x + 3 y^{2})^{7}

Aplicar o teorema do binômio:

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

a = 5 x, b = 3 y^{2}

= i = 0 \sum 7 (i 7) (5 x)^{(7 - i)} (3 y^{2})^{i}

Expandir a soma

= \frac{7 !}{0 ! ( 7 - 0 ) !} (5 x)^{7} (3 y^{2})^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} (5 x)^{6} (3 y^{2})^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} (5 x)^{5} (3 y^{2})^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} (5 x)^{4} (3 y^{2})^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} (5 x)^{3} (3 y^{2})^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} (5 x)^{2} (3 y^{2})^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} (5 x)^{1} (3 y^{2})^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} (5 x)^{0} (3 y^{2})^{7}

Simplificar

\frac{7 !}{0 ! ( 7 - 0 ) !} (5 x)^{7} (3 y^{2})^{0} : 78125 x^{7}

Simplificar

\frac{7 !}{1 ! ( 7 - 1 ) !} (5 x)^{6} (3 y^{2})^{1} : 328125 x^{6} y^{2}

Simplificar

\frac{7 !}{2 ! ( 7 - 2 ) !} (5 x)^{5} (3 y^{2})^{2} : 590625 x^{5} y^{4}

Simplificar

\frac{7 !}{3 ! ( 7 - 3 ) !} (5 x)^{4} (3 y^{2})^{3} : 590625 x^{4} y^{6}

Simplificar

\frac{7 !}{4 ! ( 7 - 4 ) !} (5 x)^{3} (3 y^{2})^{4} : 354375 x^{3} y^{8}

Simplificar

\frac{7 !}{5 ! ( 7 - 5 ) !} (5 x)^{2} (3 y^{2})^{5} : 127575 x^{2} y^{10}

Simplificar

\frac{7 !}{6 ! ( 7 - 6 ) !} (5 x)^{1} (3 y^{2})^{6} : 25515 x y^{12}

Simplificar

\frac{7 !}{7 ! ( 7 - 7 ) !} (5 x)^{0} (3 y^{2})^{7} : 2187 y^{14}

= 78125 x^{7} + 328125 x^{6} y^{2} + 590625 x^{5} y^{4} + 590625 x^{4} y^{6} + 354375 x^{3} y^{8} + 127575 x^{2} y^{10} + 25515 x y^{12} + 2187 y^{14}

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