What is expand (x^3-1/(2x))^{12} ?

The solution to expand (x^3-1/(2x))^{12} is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

expand (x^3-1/(2x))^{12}

expand

(x^{3} - \frac{1}{2 x})^{12}

x^{36} - 6 x^{32} + \frac{33 x ^{28}}{2} - \frac{55 x ^{24}}{2} + \frac{495 x ^{20}}{16} - \frac{99 x ^{16}}{4} + \frac{231 x ^{12}}{16} - \frac{99 x ^{8}}{16} + \frac{495 x ^{4}}{256} - \frac{55}{128} + \frac{33}{512 x ^{4}} - \frac{3}{512 x ^{8}} + \frac{1}{4096 x ^{12}}

Solution steps

(x^{3} - \frac{1}{2 x})^{12}

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 12 (i 12) (x^{3})^{(12 - i)} (- \frac{1}{2 x})^{i}

Expand summation

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

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} (x^{3})^{12} (- \frac{1}{2 x})^{0} : x^{36}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (x^{3})^{11} (- \frac{1}{2 x})^{1} : - 6 x^{32}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (x^{3})^{10} (- \frac{1}{2 x})^{2} : \frac{33 x ^{28}}{2}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (x^{3})^{9} (- \frac{1}{2 x})^{3} : - \frac{55 x ^{24}}{2}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (x^{3})^{8} (- \frac{1}{2 x})^{4} : \frac{495 x ^{20}}{16}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (x^{3})^{7} (- \frac{1}{2 x})^{5} : - \frac{99 x ^{16}}{4}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (x^{3})^{6} (- \frac{1}{2 x})^{6} : \frac{231 x ^{12}}{16}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (x^{3})^{5} (- \frac{1}{2 x})^{7} : - \frac{99 x ^{8}}{16}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (x^{3})^{4} (- \frac{1}{2 x})^{8} : \frac{495 x ^{4}}{256}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (x^{3})^{3} (- \frac{1}{2 x})^{9} : - \frac{55}{128}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (x^{3})^{2} (- \frac{1}{2 x})^{10} : \frac{33}{512 x ^{4}}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (x^{3})^{1} (- \frac{1}{2 x})^{11} : - \frac{3}{512 x ^{8}}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (x^{3})^{0} (- \frac{1}{2 x})^{12} : \frac{1}{4096 x ^{12}}

= x^{36} - 6 x^{32} + \frac{33 x ^{28}}{2} - \frac{55 x ^{24}}{2} + \frac{495 x ^{20}}{16} - \frac{99 x ^{16}}{4} + \frac{231 x ^{12}}{16} - \frac{99 x ^{8}}{16} + \frac{495 x ^{4}}{256} - \frac{55}{128} + \frac{33}{512 x ^{4}} - \frac{3}{512 x ^{8}} + \frac{1}{4096 x ^{12}}

(3 - x 3)^{2}

cos (x) + ln (y) + (\frac{x}{y} + e^{y}) \frac{d y}{d x}

\frac{( x - 8 ) \cdot ( x - 9 )}{30}

\frac{( ( k + 1 ) ^{2} ( k + 2 ) ^{2} )}{4}

(e^{- 5 t} + 2)^{2}