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

 Apr 15, 2022

Best Answer 

 #1
avatar+326 
+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

 

smileysmileysmiley

 Apr 15, 2022
 #1
avatar+326 
+3
Best Answer

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

 

smileysmileysmiley

Vinculum Apr 15, 2022
 #2
avatar+122390 
+1

THX, Vinculum......this one is a little  tricky  !!!

 

cool cool cool

CPhill  Apr 15, 2022

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