What is expand (x^{1/2}+x^{2/3})^9 ?

The solution to expand (x^{1/2}+x^{2/3})^9 is

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expand (x^{1/2}+x^{2/3})^9

expand

(x^{\frac{1}{2}} + x^{\frac{2}{3}})^{9}

x^{\frac{9}{2}} + 9 x^{\frac{14}{3}} + 36 x^{\frac{29}{6}} + 84 x^{5} + 126 x^{\frac{31}{6}} + 126 x^{\frac{16}{3}} + 84 x^{\frac{11}{2}} + 36 x^{\frac{17}{3}} + 9 x^{\frac{35}{6}} + x^{6}

Solution steps

(x^{\frac{1}{2}} + x^{\frac{2}{3}})^{9}

Apply binomial theorem:

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

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

= i = 0 \sum 9 (i 9) (x^{\frac{1}{2}})^{(9 - i)} (x^{\frac{2}{3}})^{i}

Expand summation

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

Simplify

\frac{9 !}{0 ! ( 9 - 0 ) !} (x^{\frac{1}{2}})^{9} (x^{\frac{2}{3}})^{0} : x^{\frac{9}{2}}

Simplify

\frac{9 !}{1 ! ( 9 - 1 ) !} (x^{\frac{1}{2}})^{8} (x^{\frac{2}{3}})^{1} : 9 x^{\frac{14}{3}}

Simplify

\frac{9 !}{2 ! ( 9 - 2 ) !} (x^{\frac{1}{2}})^{7} (x^{\frac{2}{3}})^{2} : 36 x^{\frac{29}{6}}

Simplify

\frac{9 !}{3 ! ( 9 - 3 ) !} (x^{\frac{1}{2}})^{6} (x^{\frac{2}{3}})^{3} : 84 x^{5}

Simplify

\frac{9 !}{4 ! ( 9 - 4 ) !} (x^{\frac{1}{2}})^{5} (x^{\frac{2}{3}})^{4} : 126 x^{\frac{31}{6}}

Simplify

\frac{9 !}{5 ! ( 9 - 5 ) !} (x^{\frac{1}{2}})^{4} (x^{\frac{2}{3}})^{5} : 126 x^{\frac{16}{3}}

Simplify

\frac{9 !}{6 ! ( 9 - 6 ) !} (x^{\frac{1}{2}})^{3} (x^{\frac{2}{3}})^{6} : 84 x^{\frac{11}{2}}

Simplify

\frac{9 !}{7 ! ( 9 - 7 ) !} (x^{\frac{1}{2}})^{2} (x^{\frac{2}{3}})^{7} : 36 x^{\frac{17}{3}}

Simplify

\frac{9 !}{8 ! ( 9 - 8 ) !} (x^{\frac{1}{2}})^{1} (x^{\frac{2}{3}})^{8} : 9 x^{\frac{35}{6}}

Simplify

\frac{9 !}{9 ! ( 9 - 9 ) !} (x^{\frac{1}{2}})^{0} (x^{\frac{2}{3}})^{9} : x^{6}

= x^{\frac{9}{2}} + 9 x^{\frac{14}{3}} + 36 x^{\frac{29}{6}} + 84 x^{5} + 126 x^{\frac{31}{6}} + 126 x^{\frac{16}{3}} + 84 x^{\frac{11}{2}} + 36 x^{\frac{17}{3}} + 9 x^{\frac{35}{6}} + x^{6}

a (4, - 5) b (10, 4)

(x + 5) (x - 7)

\frac{x ( s + 2 )}{s ^{3} + 10 s ^{2}}

(a + 3) (a - 3)

\frac{7 - x ^{5}}{3}

What is expand (x^{1/2}+x^{2/3})^9 ?
The solution to expand (x^{1/2}+x^{2/3})^9 is