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Popular Álgebra >

expandir (4x+5y)^6

Solução

expandir

(4 x + 5 y)^{6}

Solução

4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 75000 x y^{5} + 15625 y^{6}

Passos da solução

(4 x + 5 y)^{6}

Aplicar o teorema do binômio:

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

a = 4 x, b = 5 y

= i = 0 \sum 6 (i 6) (4 x)^{(6 - i)} (5 y)^{i}

Expandir a soma

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

Simplificar

\frac{6 !}{0 ! ( 6 - 0 ) !} (4 x)^{6} (5 y)^{0} : 4096 x^{6}

Simplificar

\frac{6 !}{1 ! ( 6 - 1 ) !} (4 x)^{5} (5 y)^{1} : 30720 x^{5} y

Simplificar

\frac{6 !}{2 ! ( 6 - 2 ) !} (4 x)^{4} (5 y)^{2} : 96000 x^{4} y^{2}

Simplificar

\frac{6 !}{3 ! ( 6 - 3 ) !} (4 x)^{3} (5 y)^{3} : 160000 x^{3} y^{3}

Simplificar

\frac{6 !}{4 ! ( 6 - 4 ) !} (4 x)^{2} (5 y)^{4} : 150000 x^{2} y^{4}

Simplificar

\frac{6 !}{5 ! ( 6 - 5 ) !} (4 x)^{1} (5 y)^{5} : 75000 x y^{5}

Simplificar

\frac{6 !}{6 ! ( 6 - 6 ) !} (4 x)^{0} (5 y)^{6} : 15625 y^{6}

= 4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 75000 x y^{5} + 15625 y^{6}

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