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What is the coefficient of x^3  in the expansion of (x + 2*sqrt(3))^5?

 Jun 27, 2021
 #1
avatar+139 
+3

ahh i do not really remember how to work these, but i will do what i can

 

we are asked to find the coefficent of $ x^3  $ in $ \left(x+2\sqrt{3}\right)^5 $

 

in these types of binomials, your exponent power value is going to be the $n$ and the $s$, or $i$, or $a$, i do not know what you know them as is going to be the coefficient thinge -- the one we are looking for is $i=3$

 

so we know that $n=5$ and $i=3$


plug it all into $ \frac{n!}{i!\left(n-i\right)!}x^{\left(n-i\right)}\left(2\sqrt{3}\right)^i  $

 

as following, we work it out:

 

 $\frac{5!}{3!\left(5-3\right)!}x^2\left(2\sqrt{3}\right)^3  $


$\frac{120}{6\cdot \:2}x^2\left(2\sqrt{3}\right)^3  $

 

$ 10\left(2\sqrt{3}\right)^3x^2  $

 

$ 10\cdot \: 8\cdot \: 3^{\frac{1}{2}\cdot \:3} x^2  $ 

 

$  10\cdot \: 8\cdot \: 3^{\frac{3}{2}}  x^2    $

 

$    10\cdot \: 8\cdot \: 3^{\frac{3}{2}}  x^2    $

 

$ 10\cdot \:8\cdot \:3\sqrt{3}\cdot \: x^2  $

 

$ \boxed{240\cdot \sqrt{3}\cdot \: x^2 }  $

 

if you want to get into decimals it would be:

 

$  240\cdot 1.73\cdot \: x^2   $    

 

$\boxed{ 415.2\cdot \:x^2 }  $

 

:D

 Jun 27, 2021
 #2
avatar+121000 
+3

2sqrt 3  =  =sqrt 12

 

So

 

The coefficient  is                

 

C ( 5 ,  2)   ( x)^(5-2)  (sqrt 12)^2 =

 

10  x^3  * 12    = 

 

120x^3 

 

 

cool cool cool  

 Jun 27, 2021
 #3
avatar+139 
+3

wait actually, no, i do not think thats right.

 

the cube root and the exponential thingie do not just cancel -- it would be $ 3\sqrt{3}  $ no?

UsernameTooShort  Jun 27, 2021
edited by UsernameTooShort  Jun 27, 2021
edited by UsernameTooShort  Jun 27, 2021
 #4
avatar+121000 
+2

No biggie.....I still gave you a point for a  good  effort  !!!!

 

 

cool cool cool

CPhill  Jun 27, 2021
 #5
avatar+121000 
+2

See here  :   https://www.wolframalpha.com/input/?i=%28x++%2B+sqrt+12%29%5E5

 

 

 

cool cool cool

CPhill  Jun 27, 2021
 #6
avatar+139 
+2

well, we both are talking about different things -- though yeah it is 120x^3 as its in the power of 3, not two -- good catch 👍:D

UsernameTooShort  Jun 27, 2021

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