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# Binomial Theorem

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What is the constant term in the expansion of $(x+x^{-3/2} )^{15}$?

Apr 15, 2022

#1
+3

The constant term of $$\left(x+x^{-\frac{3}{2}}\right)^{15}$$ when expanded is $$5005$$.

When the above expression is expanded, we get

Applying Binomial Therom:

$$\sum _{i=0}^{15}\binom{15}{i}x^{\left(15-i\right)}\left(x^{-\frac{3}{2}}\right)^i$$

$$x^{15}+15x^{\frac{25}{2}}+105x^{10}+455x^{\frac{15}{2}}+1365x^5+3003x^{\frac{5}{2}}+5005+\frac{6435}{x^{\frac{5}{2}}}+\frac{6435}{x^5}+\frac{5005}{x^{\frac{15}{2}}}+\frac{3003}{x^{10}}+\frac{1365}{x^{\frac{25}{2}}}+\frac{455}{x^{15}}+\frac{105}{x^{\frac{35}{2}}}+\frac{15}{x^{20}}+\frac{1}{x^{\frac{45}{2}}}$$

The constant term as seen above is $$5005$$.

-Vinculum   Apr 15, 2022

#1
+3

The constant term of $$\left(x+x^{-\frac{3}{2}}\right)^{15}$$ when expanded is $$5005$$.

When the above expression is expanded, we get

Applying Binomial Therom:

$$\sum _{i=0}^{15}\binom{15}{i}x^{\left(15-i\right)}\left(x^{-\frac{3}{2}}\right)^i$$

$$x^{15}+15x^{\frac{25}{2}}+105x^{10}+455x^{\frac{15}{2}}+1365x^5+3003x^{\frac{5}{2}}+5005+\frac{6435}{x^{\frac{5}{2}}}+\frac{6435}{x^5}+\frac{5005}{x^{\frac{15}{2}}}+\frac{3003}{x^{10}}+\frac{1365}{x^{\frac{25}{2}}}+\frac{455}{x^{15}}+\frac{105}{x^{\frac{35}{2}}}+\frac{15}{x^{20}}+\frac{1}{x^{\frac{45}{2}}}$$

The constant term as seen above is $$5005$$.

-Vinculum   Vinculum Apr 15, 2022
#2
+1

THX, Vinculum......this one is a little  tricky  !!!   CPhill  Apr 15, 2022