Can we prove these bounds for this fraction involving surds?

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2. 1. Express \[\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\] as a fraction having a rational denominator.

The question asks us to get rid of the square roots in the denominator; what method might we use?

Can we multiply \(\sqrt{a}-\sqrt{b}\) by something so that we lose the square roots?

2. 2. Show that, if \(n\) is a positive integer, then \(n(n+2)\) lies between \(n^2\) and \((n+1)^2\).

What is \(n(n+2) - (n+1)^2\)?

2. 3. If \(n\) is a positive integer, use the results in (i) and (ii) to find in terms of \(n\) two consecutive integers between which \[\frac{\sqrt{n+2}+\sqrt{n}}{\sqrt{n+2}-\sqrt{n}}\] must lie.

The question tells us to use results we’ve already proved… like many exam questions!