What is expand (x^2+1/(2x))^{11} ?

The solution to expand (x^2+1/(2x))^{11} is

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expand (x^2+1/(2x))^{11}

expand

(x^{2} + \frac{1}{2 x})^{11}

x^{22} + \frac{11 x ^{19}}{2} + \frac{55 x ^{16}}{4} + \frac{165 x ^{13}}{8} + \frac{165 x ^{10}}{8} + \frac{231 x ^{7}}{16} + \frac{231 x ^{4}}{32} + \frac{165 x}{64} + \frac{165}{256 x ^{2}} + \frac{55}{512 x ^{5}} + \frac{11}{1024 x ^{8}} + \frac{1}{2048 x ^{11}}

Solution steps

(x^{2} + \frac{1}{2 x})^{11}

Apply binomial theorem:

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

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

= i = 0 \sum 11 (i 11) (x^{2})^{(11 - i)} (\frac{1}{2 x})^{i}

Expand summation

= \frac{11 !}{0 ! ( 11 - 0 ) !} (x^{2})^{11} (\frac{1}{2 x})^{0} + \frac{11 !}{1 ! ( 11 - 1 ) !} (x^{2})^{10} (\frac{1}{2 x})^{1} + \frac{11 !}{2 ! ( 11 - 2 ) !} (x^{2})^{9} (\frac{1}{2 x})^{2} + \frac{11 !}{3 ! ( 11 - 3 ) !} (x^{2})^{8} (\frac{1}{2 x})^{3} + \frac{11 !}{4 ! ( 11 - 4 ) !} (x^{2})^{7} (\frac{1}{2 x})^{4} + \frac{11 !}{5 ! ( 11 - 5 ) !} (x^{2})^{6} (\frac{1}{2 x})^{5} + \frac{11 !}{6 ! ( 11 - 6 ) !} (x^{2})^{5} (\frac{1}{2 x})^{6} + \frac{11 !}{7 ! ( 11 - 7 ) !} (x^{2})^{4} (\frac{1}{2 x})^{7} + \frac{11 !}{8 ! ( 11 - 8 ) !} (x^{2})^{3} (\frac{1}{2 x})^{8} + \frac{11 !}{9 ! ( 11 - 9 ) !} (x^{2})^{2} (\frac{1}{2 x})^{9} + \frac{11 !}{10 ! ( 11 - 10 ) !} (x^{2})^{1} (\frac{1}{2 x})^{10} + \frac{11 !}{11 ! ( 11 - 11 ) !} (x^{2})^{0} (\frac{1}{2 x})^{11}

Simplify

\frac{11 !}{0 ! ( 11 - 0 ) !} (x^{2})^{11} (\frac{1}{2 x})^{0} : x^{22}

Simplify

\frac{11 !}{1 ! ( 11 - 1 ) !} (x^{2})^{10} (\frac{1}{2 x})^{1} : \frac{11 x ^{19}}{2}

Simplify

\frac{11 !}{2 ! ( 11 - 2 ) !} (x^{2})^{9} (\frac{1}{2 x})^{2} : \frac{55 x ^{16}}{4}

Simplify

\frac{11 !}{3 ! ( 11 - 3 ) !} (x^{2})^{8} (\frac{1}{2 x})^{3} : \frac{165 x ^{13}}{8}

Simplify

\frac{11 !}{4 ! ( 11 - 4 ) !} (x^{2})^{7} (\frac{1}{2 x})^{4} : \frac{165 x ^{10}}{8}

Simplify

\frac{11 !}{5 ! ( 11 - 5 ) !} (x^{2})^{6} (\frac{1}{2 x})^{5} : \frac{231 x ^{7}}{16}

Simplify

\frac{11 !}{6 ! ( 11 - 6 ) !} (x^{2})^{5} (\frac{1}{2 x})^{6} : \frac{231 x ^{4}}{32}

Simplify

\frac{11 !}{7 ! ( 11 - 7 ) !} (x^{2})^{4} (\frac{1}{2 x})^{7} : \frac{165 x}{64}

Simplify

\frac{11 !}{8 ! ( 11 - 8 ) !} (x^{2})^{3} (\frac{1}{2 x})^{8} : \frac{165}{256 x ^{2}}

Simplify

\frac{11 !}{9 ! ( 11 - 9 ) !} (x^{2})^{2} (\frac{1}{2 x})^{9} : \frac{55}{512 x ^{5}}

Simplify

\frac{11 !}{10 ! ( 11 - 10 ) !} (x^{2})^{1} (\frac{1}{2 x})^{10} : \frac{11}{1024 x ^{8}}

Simplify

\frac{11 !}{11 ! ( 11 - 11 ) !} (x^{2})^{0} (\frac{1}{2 x})^{11} : \frac{1}{2048 x ^{11}}

= x^{22} + \frac{11 x ^{19}}{2} + \frac{55 x ^{16}}{4} + \frac{165 x ^{13}}{8} + \frac{165 x ^{10}}{8} + \frac{231 x ^{7}}{16} + \frac{231 x ^{4}}{32} + \frac{165 x}{64} + \frac{165}{256 x ^{2}} + \frac{55}{512 x ^{5}} + \frac{11}{1024 x ^{8}} + \frac{1}{2048 x ^{11}}

\frac{1 - 2 ( a + h ) ^{2} - 1 - 2 a ^{2}}{h}

(f \circ g) (\frac{f}{- x ^{2}} + 1) (\frac{g}{- x} + 1)

4 [(6 y^{8} - 2 y^{5} + 4 y^{3} - y + 8) (- 9 x^{3} + 7 x^{2} - 2 x + 5)]

(4 + 4 sin (θ)) \cdot 4 cos (θ)

\frac{12 t ^{2} ( 5 - t )}{625}