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Popular Álgebra >

expandir (2y^2+1/y)^9

Solução

expandir

(2 y^{2} + \frac{1}{y})^{9}

Solução

512 y^{18} + 2304 y^{15} + 4608 y^{12} + 5376 y^{9} + 4032 y^{6} + 2016 y^{3} + 672 + \frac{144}{y ^{3}} + \frac{18}{y ^{6}} + \frac{1}{y ^{9}}

Passos da solução

(2 y^{2} + \frac{1}{y})^{9}

Aplicar o teorema do binômio:

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

a = 2 y^{2}, b = \frac{1}{y}

= i = 0 \sum 9 (i 9) (2 y^{2})^{(9 - i)} (\frac{1}{y})^{i}

Expandir a soma

= \frac{9 !}{0 ! ( 9 - 0 ) !} (2 y^{2})^{9} (\frac{1}{y})^{0} + \frac{9 !}{1 ! ( 9 - 1 ) !} (2 y^{2})^{8} (\frac{1}{y})^{1} + \frac{9 !}{2 ! ( 9 - 2 ) !} (2 y^{2})^{7} (\frac{1}{y})^{2} + \frac{9 !}{3 ! ( 9 - 3 ) !} (2 y^{2})^{6} (\frac{1}{y})^{3} + \frac{9 !}{4 ! ( 9 - 4 ) !} (2 y^{2})^{5} (\frac{1}{y})^{4} + \frac{9 !}{5 ! ( 9 - 5 ) !} (2 y^{2})^{4} (\frac{1}{y})^{5} + \frac{9 !}{6 ! ( 9 - 6 ) !} (2 y^{2})^{3} (\frac{1}{y})^{6} + \frac{9 !}{7 ! ( 9 - 7 ) !} (2 y^{2})^{2} (\frac{1}{y})^{7} + \frac{9 !}{8 ! ( 9 - 8 ) !} (2 y^{2})^{1} (\frac{1}{y})^{8} + \frac{9 !}{9 ! ( 9 - 9 ) !} (2 y^{2})^{0} (\frac{1}{y})^{9}

Simplificar

\frac{9 !}{0 ! ( 9 - 0 ) !} (2 y^{2})^{9} (\frac{1}{y})^{0} : 512 y^{18}

Simplificar

\frac{9 !}{1 ! ( 9 - 1 ) !} (2 y^{2})^{8} (\frac{1}{y})^{1} : 2304 y^{15}

Simplificar

\frac{9 !}{2 ! ( 9 - 2 ) !} (2 y^{2})^{7} (\frac{1}{y})^{2} : 4608 y^{12}

Simplificar

\frac{9 !}{3 ! ( 9 - 3 ) !} (2 y^{2})^{6} (\frac{1}{y})^{3} : 5376 y^{9}

Simplificar

\frac{9 !}{4 ! ( 9 - 4 ) !} (2 y^{2})^{5} (\frac{1}{y})^{4} : 4032 y^{6}

Simplificar

\frac{9 !}{5 ! ( 9 - 5 ) !} (2 y^{2})^{4} (\frac{1}{y})^{5} : 2016 y^{3}

Simplificar

\frac{9 !}{6 ! ( 9 - 6 ) !} (2 y^{2})^{3} (\frac{1}{y})^{6} : 672

Simplificar

\frac{9 !}{7 ! ( 9 - 7 ) !} (2 y^{2})^{2} (\frac{1}{y})^{7} : \frac{144}{y ^{3}}

Simplificar

\frac{9 !}{8 ! ( 9 - 8 ) !} (2 y^{2})^{1} (\frac{1}{y})^{8} : \frac{18}{y ^{6}}

Simplificar

\frac{9 !}{9 ! ( 9 - 9 ) !} (2 y^{2})^{0} (\frac{1}{y})^{9} : \frac{1}{y ^{9}}

= 512 y^{18} + 2304 y^{15} + 4608 y^{12} + 5376 y^{9} + 4032 y^{6} + 2016 y^{3} + 672 + \frac{144}{y ^{3}} + \frac{18}{y ^{6}} + \frac{1}{y ^{9}}

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