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

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

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

expand

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

\frac{x ^{27} y ^{18}}{512} - \frac{9 x ^{24} y ^{17}}{64} + \frac{9 x ^{21} y ^{16}}{2} - 84 x^{18} y^{15} + 1008 x^{15} y^{14} - 8064 x^{12} y^{13} + 43008 x^{9} y^{12} - 147456 x^{6} y^{11} + 294912 x^{3} y^{10} - 262144 y^{9}

Solution steps

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

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

Expand summation

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

Simplify

\frac{9 !}{0 ! ( 9 - 0 ) !} (\frac{1}{2} x^{3} y^{2})^{9} (- 4 y)^{0} : \frac{x ^{27} y ^{18}}{512}

Simplify

\frac{9 !}{1 ! ( 9 - 1 ) !} (\frac{1}{2} x^{3} y^{2})^{8} (- 4 y)^{1} : - \frac{9 x ^{24} y ^{17}}{64}

Simplify

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

Simplify

\frac{9 !}{3 ! ( 9 - 3 ) !} (\frac{1}{2} x^{3} y^{2})^{6} (- 4 y)^{3} : - 84 x^{18} y^{15}

Simplify

\frac{9 !}{4 ! ( 9 - 4 ) !} (\frac{1}{2} x^{3} y^{2})^{5} (- 4 y)^{4} : 1008 x^{15} y^{14}

Simplify

\frac{9 !}{5 ! ( 9 - 5 ) !} (\frac{1}{2} x^{3} y^{2})^{4} (- 4 y)^{5} : - 8064 x^{12} y^{13}

Simplify

\frac{9 !}{6 ! ( 9 - 6 ) !} (\frac{1}{2} x^{3} y^{2})^{3} (- 4 y)^{6} : 43008 x^{9} y^{12}

Simplify

\frac{9 !}{7 ! ( 9 - 7 ) !} (\frac{1}{2} x^{3} y^{2})^{2} (- 4 y)^{7} : - 147456 x^{6} y^{11}

Simplify

\frac{9 !}{8 ! ( 9 - 8 ) !} (\frac{1}{2} x^{3} y^{2})^{1} (- 4 y)^{8} : 294912 x^{3} y^{10}

Simplify

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

= \frac{x ^{27} y ^{18}}{512} - \frac{9 x ^{24} y ^{17}}{64} + \frac{9 x ^{21} y ^{16}}{2} - 84 x^{18} y^{15} + 1008 x^{15} y^{14} - 8064 x^{12} y^{13} + 43008 x^{9} y^{12} - 147456 x^{6} y^{11} + 294912 x^{3} y^{10} - 262144 y^{9}

A (- 2, 5) B (7, - 3)

(4 x^{\frac{1}{2}} + 6 x^{\frac{3}{2}}) (3 x - 9)

\frac{2 ( 2 z + 1 )}{4 z + 1}

(3 - x) (\frac{1}{3} + \frac{2}{27} x^{2} + \frac{2}{81} x^{3})

\frac{2 ( 6 + 5 )}{3}