What is expand (x^3+1/(2x))^{15} ?

The solution to expand (x^3+1/(2x))^{15} is

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expand (x^3+1/(2x))^{15}

expand

(x^{3} + \frac{1}{2 x})^{15}

x^{45} + \frac{15 x ^{41}}{2} + \frac{105 x ^{37}}{4} + \frac{455 x ^{33}}{8} + \frac{1365 x ^{29}}{16} + \frac{3003 x ^{25}}{32} + \frac{5005 x ^{21}}{64} + \frac{6435 x ^{17}}{128} + \frac{6435 x ^{13}}{256} + \frac{5005 x ^{9}}{512} + \frac{3003 x ^{5}}{1024} + \frac{1365 x}{2048} + \frac{455}{4096 x ^{3}} + \frac{105}{8192 x ^{7}} + \frac{15}{16384 x ^{11}} + \frac{1}{32768 x ^{15}}

Solution steps

(x^{3} + \frac{1}{2 x})^{15}

Apply binomial theorem:

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

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

= i = 0 \sum 15 (i 15) (x^{3})^{(15 - i)} (\frac{1}{2 x})^{i}

Expand summation

= \frac{15 !}{0 ! ( 15 - 0 ) !} (x^{3})^{15} (\frac{1}{2 x})^{0} + \frac{15 !}{1 ! ( 15 - 1 ) !} (x^{3})^{14} (\frac{1}{2 x})^{1} + \frac{15 !}{2 ! ( 15 - 2 ) !} (x^{3})^{13} (\frac{1}{2 x})^{2} + \frac{15 !}{3 ! ( 15 - 3 ) !} (x^{3})^{12} (\frac{1}{2 x})^{3} + \frac{15 !}{4 ! ( 15 - 4 ) !} (x^{3})^{11} (\frac{1}{2 x})^{4} + \frac{15 !}{5 ! ( 15 - 5 ) !} (x^{3})^{10} (\frac{1}{2 x})^{5} + \frac{15 !}{6 ! ( 15 - 6 ) !} (x^{3})^{9} (\frac{1}{2 x})^{6} + \frac{15 !}{7 ! ( 15 - 7 ) !} (x^{3})^{8} (\frac{1}{2 x})^{7} + \frac{15 !}{8 ! ( 15 - 8 ) !} (x^{3})^{7} (\frac{1}{2 x})^{8} + \frac{15 !}{9 ! ( 15 - 9 ) !} (x^{3})^{6} (\frac{1}{2 x})^{9} + \frac{15 !}{10 ! ( 15 - 10 ) !} (x^{3})^{5} (\frac{1}{2 x})^{10} + \frac{15 !}{11 ! ( 15 - 11 ) !} (x^{3})^{4} (\frac{1}{2 x})^{11} + \frac{15 !}{12 ! ( 15 - 12 ) !} (x^{3})^{3} (\frac{1}{2 x})^{12} + \frac{15 !}{13 ! ( 15 - 13 ) !} (x^{3})^{2} (\frac{1}{2 x})^{13} + \frac{15 !}{14 ! ( 15 - 14 ) !} (x^{3})^{1} (\frac{1}{2 x})^{14} + \frac{15 !}{15 ! ( 15 - 15 ) !} (x^{3})^{0} (\frac{1}{2 x})^{15}

Simplify

\frac{15 !}{0 ! ( 15 - 0 ) !} (x^{3})^{15} (\frac{1}{2 x})^{0} : x^{45}

Simplify

\frac{15 !}{1 ! ( 15 - 1 ) !} (x^{3})^{14} (\frac{1}{2 x})^{1} : \frac{15 x ^{41}}{2}

Simplify

\frac{15 !}{2 ! ( 15 - 2 ) !} (x^{3})^{13} (\frac{1}{2 x})^{2} : \frac{105 x ^{37}}{4}

Simplify

\frac{15 !}{3 ! ( 15 - 3 ) !} (x^{3})^{12} (\frac{1}{2 x})^{3} : \frac{455 x ^{33}}{8}

Simplify

\frac{15 !}{4 ! ( 15 - 4 ) !} (x^{3})^{11} (\frac{1}{2 x})^{4} : \frac{1365 x ^{29}}{16}

Simplify

\frac{15 !}{5 ! ( 15 - 5 ) !} (x^{3})^{10} (\frac{1}{2 x})^{5} : \frac{3003 x ^{25}}{32}

Simplify

\frac{15 !}{6 ! ( 15 - 6 ) !} (x^{3})^{9} (\frac{1}{2 x})^{6} : \frac{5005 x ^{21}}{64}

Simplify

\frac{15 !}{7 ! ( 15 - 7 ) !} (x^{3})^{8} (\frac{1}{2 x})^{7} : \frac{6435 x ^{17}}{128}

Simplify

\frac{15 !}{8 ! ( 15 - 8 ) !} (x^{3})^{7} (\frac{1}{2 x})^{8} : \frac{6435 x ^{13}}{256}

Simplify

\frac{15 !}{9 ! ( 15 - 9 ) !} (x^{3})^{6} (\frac{1}{2 x})^{9} : \frac{5005 x ^{9}}{512}

Simplify

\frac{15 !}{10 ! ( 15 - 10 ) !} (x^{3})^{5} (\frac{1}{2 x})^{10} : \frac{3003 x ^{5}}{1024}

Simplify

\frac{15 !}{11 ! ( 15 - 11 ) !} (x^{3})^{4} (\frac{1}{2 x})^{11} : \frac{1365 x}{2048}

Simplify

\frac{15 !}{12 ! ( 15 - 12 ) !} (x^{3})^{3} (\frac{1}{2 x})^{12} : \frac{455}{4096 x ^{3}}

Simplify

\frac{15 !}{13 ! ( 15 - 13 ) !} (x^{3})^{2} (\frac{1}{2 x})^{13} : \frac{105}{8192 x ^{7}}

Simplify

\frac{15 !}{14 ! ( 15 - 14 ) !} (x^{3})^{1} (\frac{1}{2 x})^{14} : \frac{15}{16384 x ^{11}}

Simplify

\frac{15 !}{15 ! ( 15 - 15 ) !} (x^{3})^{0} (\frac{1}{2 x})^{15} : \frac{1}{32768 x ^{15}}

= x^{45} + \frac{15 x ^{41}}{2} + \frac{105 x ^{37}}{4} + \frac{455 x ^{33}}{8} + \frac{1365 x ^{29}}{16} + \frac{3003 x ^{25}}{32} + \frac{5005 x ^{21}}{64} + \frac{6435 x ^{17}}{128} + \frac{6435 x ^{13}}{256} + \frac{5005 x ^{9}}{512} + \frac{3003 x ^{5}}{1024} + \frac{1365 x}{2048} + \frac{455}{4096 x ^{3}} + \frac{105}{8192 x ^{7}} + \frac{15}{16384 x ^{11}} + \frac{1}{32768 x ^{15}}

\frac{2}{s ( s + 0.104 )}

\frac{( 2 c - 10 ) ^{2}}{112} + 50

(m + 3) (m - 3)

\frac{21}{( x + 7 ) ^{2}}