What is expand (2t+1/(t^2))^{10} ?

The solution to expand (2t+1/(t^2))^{10} is

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expand (2t+1/(t^2))^{10}

expand

(2 t + \frac{1}{t ^{2}})^{10}

1024 t^{10} + 5120 t^{7} + 11520 t^{4} + 15360 t + \frac{13440}{t ^{2}} + \frac{8064}{t ^{5}} + \frac{3360}{t ^{8}} + \frac{960}{t ^{11}} + \frac{180}{t ^{14}} + \frac{20}{t ^{17}} + \frac{1}{t ^{20}}

Solution steps

(2 t + \frac{1}{t ^{2}})^{10}

Apply binomial theorem:

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

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

= i = 0 \sum 10 (i 10) (2 t)^{(10 - i)} (\frac{1}{t ^{2}})^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} (2 t)^{10} (\frac{1}{t ^{2}})^{0} : 1024 t^{10}

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} (2 t)^{9} (\frac{1}{t ^{2}})^{1} : 5120 t^{7}

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} (2 t)^{8} (\frac{1}{t ^{2}})^{2} : 11520 t^{4}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} (2 t)^{7} (\frac{1}{t ^{2}})^{3} : 15360 t

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} (2 t)^{6} (\frac{1}{t ^{2}})^{4} : \frac{13440}{t ^{2}}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} (2 t)^{5} (\frac{1}{t ^{2}})^{5} : \frac{8064}{t ^{5}}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} (2 t)^{4} (\frac{1}{t ^{2}})^{6} : \frac{3360}{t ^{8}}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} (2 t)^{3} (\frac{1}{t ^{2}})^{7} : \frac{960}{t ^{11}}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} (2 t)^{2} (\frac{1}{t ^{2}})^{8} : \frac{180}{t ^{14}}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} (2 t)^{1} (\frac{1}{t ^{2}})^{9} : \frac{20}{t ^{17}}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} (2 t)^{0} (\frac{1}{t ^{2}})^{10} : \frac{1}{t ^{20}}

= 1024 t^{10} + 5120 t^{7} + 11520 t^{4} + 15360 t + \frac{13440}{t ^{2}} + \frac{8064}{t ^{5}} + \frac{3360}{t ^{8}} + \frac{960}{t ^{11}} + \frac{180}{t ^{14}} + \frac{20}{t ^{17}} + \frac{1}{t ^{20}}

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