#1**+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 ]

CPhill Jul 23, 2019

#2**0 **

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

ElectricPavlov Jul 23, 2019

#3**+1 **

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

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

heureka Jul 24, 2019