How to simplify 4+265+26I was tackling through an olympiad practice book when I saw one of these problems:If x=5+26, evaluate x−1xThe answer written is 22, but I can't figure my way out through the manipulations. I just know that I have the following:4+265+26

Hint Squaring it, we get 40+1665+26=8Remark Generally n−1+n2−1n+n2−1=2(n−1) with analogous proof.For another example note 3+1110+99=2, by n=10 above.

Resolved: "How to simplify 4+265+26I was tackling through an..."

Delilah Novak

2022-04-07

How to simplify

\frac{4 + 2 \sqrt{6}}{\sqrt{5 + 2 \sqrt{6}}}

I was tackling through an olympiad practice book when I saw one of these problems:
If

x = 5 + 2 \sqrt{6}

, evaluate

\frac{x - 1}{\sqrt{x}}

The answer written is

2 \sqrt{2}

, but I can't figure my way out through the manipulations. I just know that I have the following:

\frac{4 + 2 \sqrt{6}}{\sqrt{5 + 2 \sqrt{6}}}

Emily Green

Beginner2022-04-08Added 14 answers

Hint Squaring it, we get

\frac{40 + 16 \sqrt{6}}{5 + 2 \sqrt{6}} = 8

Remark Generally

\frac{n - 1 + \sqrt{n^{2} - 1}}{\sqrt{n + \sqrt{n^{2} - 1}}} = \sqrt{2 (n - 1)}

with analogous proof.
For another example note

\frac{3 + \sqrt{11}}{\sqrt{10 + \sqrt{99}}} = \sqrt{2}

, by

n = 10

above.

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