What is expand (1/3 x^3y^2-4y)^9 ?

The solution to expand (1/3 x^3y^2-4y)^9 is

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expand (1/3 x^3y^2-4y)^9

expand

(\frac{1}{3} x^{3} y^{2} - 4 y)^{9}

\frac{x ^{27} y ^{18}}{19683} - \frac{4 x ^{24} y ^{17}}{729} + \frac{64 x ^{21} y ^{16} - 1792 x ^{18} y ^{15}}{243} + \frac{3584 x ^{15} y ^{14}}{27} + \frac{- 14336 x ^{12} y ^{13} + 114688 x ^{9} y ^{12}}{9} - 65536 x^{6} y^{11} + 196608 x^{3} y^{10} - 262144 y^{9}

Solution steps

(\frac{1}{3} x^{3} y^{2} - 4 y)^{9}

Apply binomial theorem:

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

a = \frac{1}{3} x^{3} y^{2}, b = - 4 y

= i = 0 \sum 9 (i 9) (\frac{1}{3} x^{3} y^{2})^{(9 - i)} (- 4 y)^{i}

Expand summation

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

Simplify

\frac{9 !}{0 ! ( 9 - 0 ) !} (\frac{1}{3} x^{3} y^{2})^{9} (- 4 y)^{0} : \frac{x ^{27} y ^{18}}{19683}

Simplify

\frac{9 !}{1 ! ( 9 - 1 ) !} (\frac{1}{3} x^{3} y^{2})^{8} (- 4 y)^{1} : - \frac{4 x ^{24} y ^{17}}{729}

Simplify

\frac{9 !}{2 ! ( 9 - 2 ) !} (\frac{1}{3} x^{3} y^{2})^{7} (- 4 y)^{2} : \frac{64 x ^{21} y ^{16}}{243}

Simplify

\frac{9 !}{3 ! ( 9 - 3 ) !} (\frac{1}{3} x^{3} y^{2})^{6} (- 4 y)^{3} : - \frac{1792 x ^{18} y ^{15}}{243}

Simplify

\frac{9 !}{4 ! ( 9 - 4 ) !} (\frac{1}{3} x^{3} y^{2})^{5} (- 4 y)^{4} : \frac{3584 x ^{15} y ^{14}}{27}

Simplify

\frac{9 !}{5 ! ( 9 - 5 ) !} (\frac{1}{3} x^{3} y^{2})^{4} (- 4 y)^{5} : - \frac{14336 x ^{12} y ^{13}}{9}

Simplify

\frac{9 !}{6 ! ( 9 - 6 ) !} (\frac{1}{3} x^{3} y^{2})^{3} (- 4 y)^{6} : \frac{114688 x ^{9} y ^{12}}{9}

Simplify

\frac{9 !}{7 ! ( 9 - 7 ) !} (\frac{1}{3} x^{3} y^{2})^{2} (- 4 y)^{7} : - 65536 x^{6} y^{11}

Simplify

\frac{9 !}{8 ! ( 9 - 8 ) !} (\frac{1}{3} x^{3} y^{2})^{1} (- 4 y)^{8} : 196608 x^{3} y^{10}

Simplify

\frac{9 !}{9 ! ( 9 - 9 ) !} (\frac{1}{3} x^{3} y^{2})^{0} (- 4 y)^{9} : - 262144 y^{9}

= \frac{x ^{27} y ^{18}}{19683} - \frac{4 x ^{24} y ^{17}}{729} + \frac{64 x ^{21} y ^{16}}{243} - \frac{1792 x ^{18} y ^{15}}{243} + \frac{3584 x ^{15} y ^{14}}{27} - \frac{14336 x ^{12} y ^{13}}{9} + \frac{114688 x ^{9} y ^{12}}{9} - 65536 x^{6} y^{11} + 196608 x^{3} y^{10} - 262144 y^{9}

Join

\frac{x ^{27} y ^{18}}{19683} - \frac{4 x ^{24} y ^{17}}{729} + \frac{64 x ^{21} y ^{16}}{243} - \frac{1792 x ^{18} y ^{15}}{243} + \frac{3584 x ^{15} y ^{14}}{27} - \frac{14336 x ^{12} y ^{13}}{9} + \frac{114688 x ^{9} y ^{12}}{9} - 65536 x^{6} y^{11} + 196608 x^{3} y^{10} - 262144 y^{9} : \frac{x ^{27} y ^{18}}{19683} - \frac{4 x ^{24} y ^{17}}{729} + \frac{64 x ^{21} y ^{16} - 1792 x ^{18} y ^{15}}{243} + \frac{3584 x ^{15} y ^{14}}{27} + \frac{- 14336 x ^{12} y ^{13} + 114688 x ^{9} y ^{12}}{9} - 65536 x^{6} y^{11} + 196608 x^{3} y^{10} - 262144 y^{9}

= \frac{x ^{27} y ^{18}}{19683} - \frac{4 x ^{24} y ^{17}}{729} + \frac{64 x ^{21} y ^{16} - 1792 x ^{18} y ^{15}}{243} + \frac{3584 x ^{15} y ^{14}}{27} + \frac{- 14336 x ^{12} y ^{13} + 114688 x ^{9} y ^{12}}{9} - 65536 x^{6} y^{11} + 196608 x^{3} y^{10} - 262144 y^{9}

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