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Popular >

expand (x/2+y^2)^{12}

Solution

expand

(\frac{x}{2} + y^{2})^{12}

Solution

\frac{x ^{12}}{4096} + \frac{3 x ^{11} y ^{2} + 33 x ^{10} y ^{4}}{512} + \frac{55 x ^{9} y ^{6}}{128} + \frac{495 x ^{8} y ^{8}}{256} + \frac{99 x ^{7} y ^{10} + 231 x ^{6} y ^{12} + 495 x ^{4} y ^{16}}{16} + \frac{99 x ^{5} y ^{14}}{4} + \frac{55 x ^{3} y ^{18} + 33 x ^{2} y ^{20}}{2} + 6 x y^{22} + y^{24}

Solution steps

(\frac{x}{2} + y^{2})^{12}

Apply binomial theorem:

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

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

= i = 0 \sum 12 (i 12) (\frac{x}{2})^{(12 - i)} (y^{2})^{i}

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (\frac{x}{2})^{9} (y^{2})^{3} : \frac{55 x ^{9} y ^{6}}{128}

Simplify

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

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (\frac{x}{2})^{7} (y^{2})^{5} : \frac{99 x ^{7} y ^{10}}{16}

Simplify

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

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (\frac{x}{2})^{5} (y^{2})^{7} : \frac{99 x ^{5} y ^{14}}{4}

Simplify

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

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (\frac{x}{2})^{3} (y^{2})^{9} : \frac{55 x ^{3} y ^{18}}{2}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (\frac{x}{2})^{2} (y^{2})^{10} : \frac{33 x ^{2} y ^{20}}{2}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (\frac{x}{2})^{1} (y^{2})^{11} : 6 x y^{22}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (\frac{x}{2})^{0} (y^{2})^{12} : y^{24}

= \frac{x ^{12}}{4096} + \frac{3 x ^{11} y ^{2}}{512} + \frac{33 x ^{10} y ^{4}}{512} + \frac{55 x ^{9} y ^{6}}{128} + \frac{495 x ^{8} y ^{8}}{256} + \frac{99 x ^{7} y ^{10}}{16} + \frac{231 x ^{6} y ^{12}}{16} + \frac{99 x ^{5} y ^{14}}{4} + \frac{495 x ^{4} y ^{16}}{16} + \frac{55 x ^{3} y ^{18}}{2} + \frac{33 x ^{2} y ^{20}}{2} + 6 x y^{22} + y^{24}

Join

\frac{x ^{12}}{4096} + \frac{3 x ^{11} y ^{2}}{512} + \frac{33 x ^{10} y ^{4}}{512} + \frac{55 x ^{9} y ^{6}}{128} + \frac{495 x ^{8} y ^{8}}{256} + \frac{99 x ^{7} y ^{10}}{16} + \frac{231 x ^{6} y ^{12}}{16} + \frac{99 x ^{5} y ^{14}}{4} + \frac{495 x ^{4} y ^{16}}{16} + \frac{55 x ^{3} y ^{18}}{2} + \frac{33 x ^{2} y ^{20}}{2} + 6 x y^{22} + y^{24} : \frac{x ^{12}}{4096} + \frac{3 x ^{11} y ^{2} + 33 x ^{10} y ^{4}}{512} + \frac{55 x ^{9} y ^{6}}{128} + \frac{495 x ^{8} y ^{8}}{256} + \frac{99 x ^{7} y ^{10} + 231 x ^{6} y ^{12} + 495 x ^{4} y ^{16}}{16} + \frac{99 x ^{5} y ^{14}}{4} + \frac{55 x ^{3} y ^{18} + 33 x ^{2} y ^{20}}{2} + 6 x y^{22} + y^{24}

= \frac{x ^{12}}{4096} + \frac{3 x ^{11} y ^{2} + 33 x ^{10} y ^{4}}{512} + \frac{55 x ^{9} y ^{6}}{128} + \frac{495 x ^{8} y ^{8}}{256} + \frac{99 x ^{7} y ^{10} + 231 x ^{6} y ^{12} + 495 x ^{4} y ^{16}}{16} + \frac{99 x ^{5} y ^{14}}{4} + \frac{55 x ^{3} y ^{18} + 33 x ^{2} y ^{20}}{2} + 6 x y^{22} + y^{24}

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