How do you rationalize this fraction:

$$\frac{1}{\sqrt{n+1}}$$

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possible duplicate of Rationalize the Denominator by Default – wxffles May 16 '13 at 1:13
thanks, but I'm trying to do it by hand... Is there a way to do that? – Joseph May 16 '13 at 1:22
Oh, sorry about that. I must have clicked the wrong place or was redirected incorrectly. But thanks for the help. I was over-thinking it with the conjugate things. – Joseph May 16 '13 at 1:30

To rationalize the denominator given: $$\frac{1}{\sqrt{n+1}}$$

1) Multiply both the numerator and the denominator by $\sqrt{n+1}$ $$=\frac{1}{\sqrt{n+1}}\times\frac{\sqrt{n+1}}{\sqrt{n+1}}$$

2) This leaves us with: $$\frac{\sqrt{n+1}}{\sqrt{n+1}{\sqrt{n+1}}}$$

3) Simplifying we are left with:

$$\frac{\sqrt{n+1}}{{n+1}}$$

Because:

$$\sqrt{n+1}\times\sqrt{n+1} = n+1$$

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well... you should agree that $\frac{\sqrt{n+1}}{\sqrt{n+1}}=1$, given $n\neq-1$; thus $$\frac{1}{\sqrt{n+1}}=\frac{1}{\sqrt{n+1}}\frac{\sqrt{n+1}}{\sqrt{n+1}}=\frac{\sqrt{n+1}}{\sqrt{n+1}\sqrt{n+1}}=\frac{\sqrt{n+1}}{n+1}\,.$$ Is this what you wanted?

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@Sujaan Kunalan: OK, you are right but this is clearly implicit in the question as already in the original text we are dividing by $\sqrt{n+1}$. This is why I do not find it necessary to specify this hypothesis in my answer! – Simone May 17 '13 at 9:37