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Is there a general way to write $\sqrt{a+\sqrt{b}}$ as $c + \sqrt{d}$ for $a,b,c,d$ positive integers? Is it always possible? I've seen several ways to solve specific cases like $$\sqrt{6+4\sqrt{2}} = a+\sqrt{b} $$ where you square both sides and work $a$ and $b$ out and get $a=b=2$. If it is not possible, can someone please show it for a particular case?

Watson
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omega-stable
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  • This is not possible in general. For example, $\sqrt{1+\sqrt{2}}$ cannot be reduced like that. – Alexis Olson Sep 07 '16 at 04:39
  • What conditions do $a$ and $b$ have to satisfy in order for this to be possible? – omega-stable Sep 07 '16 at 04:45
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    This is called "denesting" or "denesting radicals", there is a lot written about it, but unless you have a reason to I would suggest considering it in terms of rationals rather than integers. – DanielV Sep 07 '16 at 05:33
  • @I.Padilla : see http://math.stackexchange.com/questions/1828493, http://math.stackexchange.com/questions/196155, http://math.stackexchange.com/questions/1214527 – Watson Sep 07 '16 at 09:14

1 Answers1

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If we start with

$$\sqrt{a+\sqrt{b}} = c+\sqrt{d}$$

$$a + \sqrt{b} = c^2 + d + 2c\sqrt{d}$$

Assuming $\sqrt{b}$ is not an integer, we must have $a = c^2 + d$ and $b = 4c^2d$. We see that whenever $b$ is not of that form (e.g. not divisible by 2, for example), then we cannot reduce the original expression. (There are other times when we cannot reduce it as well, but this provides one set of counterexamples).

user2825632
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