Expanding this binomial?

Expand and simplify ${\left(\sqrt{3} + 2\right)}^{5}$. Give your answer in the from $a + b \sqrt{3}$ where a and b are whole, positive intergers.

Apr 2, 2017

${\left(\sqrt{3} + 2\right)}^{5} = 362 + 209 \sqrt{3}$

Explanation:

Let us look at what happens when $a + b \sqrt{3}$ is multiplied by $\left(\sqrt{3} + 2\right)$...

Using FOIL...

$\left(a + b \sqrt{3}\right) \left(\sqrt{3} + 2\right) = {\overbrace{a \cdot \sqrt{3}}}^{\text{First"+overbrace(a*2)^"Outside"+overbrace(bsqrt(3)*sqrt(3))^"Inside"+overbrace(bsqrt(3)*2)^"Last}}$

$\textcolor{w h i t e}{\left(a + b \sqrt{3}\right) \left(\sqrt{3} + 2\right)} = \left(2 a + 3 b\right) + \left(a + 2 b\right) \sqrt{3}$

So starting with ${a}_{0} = 1$ and ${b}_{0} = 0$, we can apply these formulae $5$ times:

$\left\{\begin{matrix}{a}_{i + 1} = 2 {a}_{i} + 3 {b}_{i} \\ {b}_{i + 1} = {a}_{i} + 2 {b}_{i}\end{matrix}\right.$

Writing $\left({a}_{i} , {b}_{i}\right)$ as a pair, we find:

$\left(1 , 0\right) \to \left(2 , 1\right) \to \left(7 , 4\right) \to \left(26 , 15\right) \to \left(97 , 56\right) \to \left(362 , 209\right)$

So:

${\left(\sqrt{3} + 2\right)}^{5} = 362 + 209 \sqrt{3}$

$\textcolor{w h i t e}{}$
Observations

$\frac{2}{1} , \frac{7}{4} , \frac{26}{15} , \frac{97}{56} , \frac{362}{209}$

are successively better rational approximations to $\sqrt{3}$

Why should this be?

Check the standard continued fraction expansion of $\sqrt{3}$:

sqrt(3) = [1;bar(1,2)] = 1+1/(1+1/(2+1/(1+1/(2+1/(1+1/(2+...))))))

The partial sums are:

$1 , \textcolor{b l u e}{\frac{2}{1}} , \frac{5}{3} , \textcolor{b l u e}{\frac{7}{4}} , \frac{19}{11} , \textcolor{b l u e}{\frac{26}{15}} , \frac{71}{41} , \textcolor{b l u e}{\frac{97}{56}} , \frac{265}{153} , \textcolor{b l u e}{\frac{362}{209}}$

The sequence of ratios we found are the (Pell equation satisfying) approximations:

[1;1] = 2/1

[1;1,2,1] = 7/4

[1;1,2,1,2,1] = 26/15

[1;1,2,1,2,1,2,1] = 97/56

[1;1,2,1,2,1,2,1,2,1] = 362/209

The rules we found for multiplying $a + b \sqrt{3}$ by $\left(\sqrt{3} + 2\right)$ are essentially the same as evaluating two steps of this continued fraction.