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What is the nearest integer to $(5+2\sqrt7)^4$?

 Jul 23, 2019
 #1
avatar+103917 
+3

\($(5+2\sqrt7)^4$ \)      using the binomial expansion

 

C(4,0) *5^4  +  C(4,1) * (5)^3 * (2√7)  +  C(4,2) * (5^2) * (2√7)^2  +  C(4, 3) * (5) (2√7)^3  + C(4, 4)*(2√7)^4  =

 

625  +  4 * (125) * 2√7  +  6 * 25 * 4*7  +  4 *5 * (8 ) *(√7)^3  + 2^4 * (√7)^4  =

 

625  +  1000√7  + 4200 +  160* 7 * √7  + 16 * 49  =

 

625 + 1000√7 + 4200 + 1120√7 + 784  =

 

5609  + 2120√7   ≈   11,218  [ to the nearest integer ]

 

 

 

cool cool cool

 Jul 23, 2019
edited by CPhill  Jul 23, 2019
 #2
avatar+19325 
+2

Using just the on-site calculator:   Is that 2 divided by sqrt 7?

(5+(2/sqrt(7)))^4 = 1097.6444772902544618   ~~ 1098

 

If it is 2 * sqrt 7:

 

(5+(2*sqrt7))^4 = 11217.9927794569320519  ~~ 11218

 Jul 23, 2019
 #3
avatar+23137 
+1

What is the nearest integer to \((5+2\sqrt7)^4\) ?

 

\(\begin{array}{|rcll|} \hline && (5+2\sqrt7)^4 \\ &=& \left(~(5+2\sqrt7)^2~\right)^2 \\ &=& \left(~25+20\sqrt7+4\cdot 7~\right)^2 \\ &=& \left(~53+20\sqrt7~\right)^2 \\ &=& \left(~53^2+2\cdot 53\cdot 20\sqrt7 + 400\cdot 7 ~\right)^2 \\ &=& 5609+2120\sqrt7 \\ &=& 5609+2120\cdot 2.64575131106 \\ &=& 5609+5608.99277946 \\ &=& 5609+5609 \quad | \quad \text{to the nearest integer } \\ &=& \mathbf{11218} \\ \hline \end{array}\)

 

laugh

 Jul 24, 2019

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