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Popular >

expand (2y-1/x)^{10}

Solution

expand

(2 y - \frac{1}{x})^{10}

Solution

1024 y^{10} - \frac{5120 y ^{9}}{x} + \frac{11520 y ^{8}}{x ^{2}} - \frac{15360 y ^{7}}{x ^{3}} + \frac{13440 y ^{6}}{x ^{4}} - \frac{8064 y ^{5}}{x ^{5}} + \frac{3360 y ^{4}}{x ^{6}} - \frac{960 y ^{3}}{x ^{7}} + \frac{180 y ^{2}}{x ^{8}} - \frac{20 y}{x ^{9}} + \frac{1}{x ^{10}}

Solution steps

(2 y - \frac{1}{x})^{10}

Apply binomial theorem:

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

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

= i = 0 \sum 10 (i 10) (2 y)^{(10 - i)} (- \frac{1}{x})^{i}

Expand summation

= \frac{10 !}{0 ! ( 10 - 0 ) !} (2 y)^{10} (- \frac{1}{x})^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} (2 y)^{9} (- \frac{1}{x})^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} (2 y)^{8} (- \frac{1}{x})^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} (2 y)^{7} (- \frac{1}{x})^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} (2 y)^{6} (- \frac{1}{x})^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} (2 y)^{5} (- \frac{1}{x})^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} (2 y)^{4} (- \frac{1}{x})^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} (2 y)^{3} (- \frac{1}{x})^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} (2 y)^{2} (- \frac{1}{x})^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} (2 y)^{1} (- \frac{1}{x})^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} (2 y)^{0} (- \frac{1}{x})^{10}

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} (2 y)^{10} (- \frac{1}{x})^{0} : 1024 y^{10}

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} (2 y)^{9} (- \frac{1}{x})^{1} : - \frac{5120 y ^{9}}{x}

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} (2 y)^{8} (- \frac{1}{x})^{2} : \frac{11520 y ^{8}}{x ^{2}}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} (2 y)^{7} (- \frac{1}{x})^{3} : - \frac{15360 y ^{7}}{x ^{3}}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} (2 y)^{6} (- \frac{1}{x})^{4} : \frac{13440 y ^{6}}{x ^{4}}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} (2 y)^{5} (- \frac{1}{x})^{5} : - \frac{8064 y ^{5}}{x ^{5}}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} (2 y)^{4} (- \frac{1}{x})^{6} : \frac{3360 y ^{4}}{x ^{6}}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} (2 y)^{3} (- \frac{1}{x})^{7} : - \frac{960 y ^{3}}{x ^{7}}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} (2 y)^{2} (- \frac{1}{x})^{8} : \frac{180 y ^{2}}{x ^{8}}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} (2 y)^{1} (- \frac{1}{x})^{9} : - \frac{20 y}{x ^{9}}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} (2 y)^{0} (- \frac{1}{x})^{10} : \frac{1}{x ^{10}}

= 1024 y^{10} - \frac{5120 y ^{9}}{x} + \frac{11520 y ^{8}}{x ^{2}} - \frac{15360 y ^{7}}{x ^{3}} + \frac{13440 y ^{6}}{x ^{4}} - \frac{8064 y ^{5}}{x ^{5}} + \frac{3360 y ^{4}}{x ^{6}} - \frac{960 y ^{3}}{x ^{7}} + \frac{180 y ^{2}}{x ^{8}} - \frac{20 y}{x ^{9}} + \frac{1}{x ^{10}}

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