Simplification of Irrational Numbers

Question

Just a quick question i'm currently confused, would be grateful if you anyone could provide full working out.

"Without resorting to the use of a calculator or computer, find a simpler representation for each of the numbers below:"

$\sqrt{2+\sqrt3} - \sqrt{2-\sqrt3}$

Extend to find: $\sqrt{2+\sqrt3} + \sqrt{2-\sqrt3}$

Generalise to find: $\sqrt{a+\sqrt b} \pm \sqrt{a-\sqrt b}$

Thanks..

I was just going to suggest some corrections but it seems that you have managed most yourself. You can use \pm for the plus / minus symbol. — badjohn, Feb 05 '18 at 10:02

score 1 · Accepted Answer · 2018-02-05T10:18:28.373

1

Notice that $2\cdot2=\sqrt3^2+1^2$ and

$$\sqrt{\frac{2(\sqrt3\pm1)^2}2}=\frac{\sqrt3\pm1}{\sqrt2}.$$

Hence your expressions are $\sqrt 6$ and $\sqrt2$.

To generalize, write

$$a\pm\sqrt b=(\alpha\pm\beta\sqrt b)^2=\alpha^2+\beta^2b\pm2\alpha\beta\sqrt b$$ and identify.

edited Feb 05 '18 at 10:18

answered Feb 05 '18 at 10:10

Thanks for the help, could you please elaborate on the generalization, still quite confused. – RalphCarroll Feb 06 '18 at 09:47
@JustyourAverage: you need to identify the $a$ and $\sqrt b$ terms and solve for $\alpha,\beta$. You will see that a condition is required to obtain a simpler solution. – Feb 06 '18 at 09:51

1 Answers1