Question

Find the third proportional to $$(2 + \sqrt{3})$$ and $$(5 + 4\sqrt{3})$$.

Answer 1

If $$x$$ is the third proportional to $$a$$ and $$b$$, then $$a : b :: b : x \Rightarrow x = \frac{b^2}{a}$$.

$$x = \frac{(5 + 4\sqrt{3})^2}{2 + \sqrt{3}}$$

$$= \frac{25 + 48 + 40\sqrt{3}}{2 + \sqrt{3}}$$

$$= \frac{73 + 40\sqrt{3}}{2 + \sqrt{3}}$$

Rationalising the denominator by multiplying numerator and denominator by $$(2 - \sqrt{3})$$:

$$x = \frac{(73 + 40\sqrt{3})(2 - \sqrt{3})}{(2)^2 - (\sqrt{3})^2}$$

$$= \frac{146 - 73\sqrt{3} + 80\sqrt{3} - 120}{4 - 3}$$

$$= 26 + 7\sqrt{3}$$