The variables \(x\) and \(y\) are connected by the relation \(y = ax^n\), where \(a\) and \(n\) are constants; \(y = 3\) when \(x = 4\) and \(y = 2\) when \(x = 9\). Find the exact values of \(n\) and \(a\).
Express \[\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\] as a fraction having a rational denominator.
Show that, if \(n\) is a positive integer, then \(n(n+2)\) lies between \(n^2\) and \((n+1)^2\).
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.
Thinking about Algebra
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