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Popular >

expand (at-t^{1/2})^9

Solution

expand

(a t - t^{\frac{1}{2}})^{9}

Solution

a^{9} t^{9} - 9 a^{8} t^{\frac{17}{2}} + 36 a^{7} t^{8} - 84 a^{6} t^{\frac{15}{2}} + 126 a^{5} t^{7} - 126 a^{4} t^{\frac{13}{2}} + 84 a^{3} t^{6} - 36 a^{2} t^{\frac{11}{2}} + 9 a t^{5} - t^{\frac{9}{2}}

Solution steps

(a t - t^{\frac{1}{2}})^{9}

Apply binomial theorem:

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

a = a t, b = - t^{\frac{1}{2}}

= i = 0 \sum 9 (i 9) (a t)^{(9 - i)} (- t^{\frac{1}{2}})^{i}

Expand summation

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

Simplify

\frac{9 !}{0 ! ( 9 - 0 ) !} (a t)^{9} (- t^{\frac{1}{2}})^{0} : a^{9} t^{9}

Simplify

\frac{9 !}{1 ! ( 9 - 1 ) !} (a t)^{8} (- t^{\frac{1}{2}})^{1} : - 9 a^{8} t^{\frac{17}{2}}

Simplify

\frac{9 !}{2 ! ( 9 - 2 ) !} (a t)^{7} (- t^{\frac{1}{2}})^{2} : 36 a^{7} t^{8}

Simplify

\frac{9 !}{3 ! ( 9 - 3 ) !} (a t)^{6} (- t^{\frac{1}{2}})^{3} : - 84 a^{6} t^{\frac{15}{2}}

Simplify

\frac{9 !}{4 ! ( 9 - 4 ) !} (a t)^{5} (- t^{\frac{1}{2}})^{4} : 126 a^{5} t^{7}

Simplify

\frac{9 !}{5 ! ( 9 - 5 ) !} (a t)^{4} (- t^{\frac{1}{2}})^{5} : - 126 a^{4} t^{\frac{13}{2}}

Simplify

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

Simplify

\frac{9 !}{7 ! ( 9 - 7 ) !} (a t)^{2} (- t^{\frac{1}{2}})^{7} : - 36 a^{2} t^{\frac{11}{2}}

Simplify

\frac{9 !}{8 ! ( 9 - 8 ) !} (a t)^{1} (- t^{\frac{1}{2}})^{8} : 9 a t^{5}

Simplify

\frac{9 !}{9 ! ( 9 - 9 ) !} (a t)^{0} (- t^{\frac{1}{2}})^{9} : - t^{\frac{9}{2}}

= a^{9} t^{9} - 9 a^{8} t^{\frac{17}{2}} + 36 a^{7} t^{8} - 84 a^{6} t^{\frac{15}{2}} + 126 a^{5} t^{7} - 126 a^{4} t^{\frac{13}{2}} + 84 a^{3} t^{6} - 36 a^{2} t^{\frac{11}{2}} + 9 a t^{5} - t^{\frac{9}{2}}

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Frequently Asked Questions (FAQ)

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