What is expand (1/2 sqrt(a)+2/3 a^{4/5})^{12} ?

The solution to expand (1/2 sqrt(a)+2/3 a^{4/5})^{12} is

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expand (1/2 sqrt(a)+2/3 a^{4/5})^{12}

expand

(\frac{1}{2} a + \frac{2}{3} a^{\frac{4}{5}})^{12}

\frac{a ^{6}}{4096} + \frac{a ^{\frac{63}{10}}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{\frac{15}{2}}}{27} + \frac{308 a ^{\frac{39}{5}} + 352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}} + 5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441}

Solution steps

(\frac{1}{2} a + \frac{2}{3} a^{\frac{4}{5}})^{12}

Apply binomial theorem:

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

a = \frac{1}{2} a, b = \frac{2}{3} a^{\frac{4}{5}}

= i = 0 \sum 12 (i 12) (\frac{1}{2} a)^{(12 - i)} (\frac{2}{3} a^{\frac{4}{5}})^{i}

Expand summation

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

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} (\frac{1}{2} a)^{12} (\frac{2}{3} a^{\frac{4}{5}})^{0} : \frac{a ^{6}}{4096}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (\frac{1}{2} a)^{11} (\frac{2}{3} a^{\frac{4}{5}})^{1} : \frac{a ^{\frac{4}{5}} ( a ) ^{11}}{256}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (\frac{1}{2} a)^{10} (\frac{2}{3} a^{\frac{4}{5}})^{2} : \frac{11 a ^{\frac{33}{5}}}{384}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (\frac{1}{2} a)^{9} (\frac{2}{3} a^{\frac{4}{5}})^{3} : \frac{55 a ^{\frac{69}{10}}}{432}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (\frac{1}{2} a)^{8} (\frac{2}{3} a^{\frac{4}{5}})^{4} : \frac{55 a ^{\frac{36}{5}}}{144}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (\frac{1}{2} a)^{7} (\frac{2}{3} a^{\frac{4}{5}})^{5} : \frac{22 a ^{4} ( a ) ^{7}}{27}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (\frac{1}{2} a)^{6} (\frac{2}{3} a^{\frac{4}{5}})^{6} : \frac{308 a ^{\frac{39}{5}}}{243}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (\frac{1}{2} a)^{5} (\frac{2}{3} a^{\frac{4}{5}})^{7} : \frac{352 a ^{\frac{81}{10}}}{243}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (\frac{1}{2} a)^{4} (\frac{2}{3} a^{\frac{4}{5}})^{8} : \frac{880 a ^{\frac{42}{5}}}{729}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (\frac{1}{2} a)^{3} (\frac{2}{3} a^{\frac{4}{5}})^{9} : \frac{14080 a ^{\frac{87}{10}}}{19683}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (\frac{1}{2} a)^{2} (\frac{2}{3} a^{\frac{4}{5}})^{10} : \frac{5632 a ^{9}}{19683}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (\frac{1}{2} a)^{1} (\frac{2}{3} a^{\frac{4}{5}})^{11} : \frac{4096 a ^{\frac{93}{10}}}{59049}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (\frac{1}{2} a)^{0} (\frac{2}{3} a^{\frac{4}{5}})^{12} : \frac{4096 a ^{\frac{48}{5}}}{531441}

= \frac{a ^{6}}{4096} + \frac{a ^{\frac{4}{5}} ( a ) ^{11}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{4} ( a ) ^{7}}{27} + \frac{308 a ^{\frac{39}{5}}}{243} + \frac{352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}}}{19683} + \frac{5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441}

Simplify

\frac{a ^{6}}{4096} + \frac{a ^{\frac{4}{5}} ( a ) ^{11}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{4} ( a ) ^{7}}{27} + \frac{308 a ^{\frac{39}{5}}}{243} + \frac{352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}}}{19683} + \frac{5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441} : \frac{a ^{6}}{4096} + \frac{a ^{\frac{63}{10}}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{\frac{15}{2}}}{27} + \frac{308 a ^{\frac{39}{5}} + 352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}} + 5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441}

= \frac{a ^{6}}{4096} + \frac{a ^{\frac{63}{10}}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{\frac{15}{2}}}{27} + \frac{308 a ^{\frac{39}{5}} + 352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}} + 5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441}

(cot (θ) + csc (θ))^{2}

8 (- x - \frac{1}{2})

a (cos (Θ) + Θ sin (Θ))

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

(7 + cos (θ))^{2}

What is expand (1/2 sqrt(a)+2/3 a^{4/5})^{12} ?
The solution to expand (1/2 sqrt(a)+2/3 a^{4/5})^{12} is