Show that $A$ is a whole number: $$A=\sqrt{\left|40\sqrt2-57\right|}-\sqrt{\left|40\sqrt2+57\right|}.$$ I don't know if this is necessary, but we can compare $40\sqrt{2}$ and $57$: $$40\sqrt{2}\Diamond57,\\1600\times2\Diamond 3249,\\3200\Diamond3249,\\3200<3249\Rightarrow 40\sqrt{2}<57.$$ Is this actually needed for the solution? So $$A=\sqrt{57-40\sqrt2}-\sqrt{40\sqrt2+57}.$$ What should I do next?
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1Just to clarify, are you using the $\Diamond$ symbol as a placeholder for $<$, $>$, $\ge$, $\le$ etc? – user170231 Apr 05 '21 at 16:48
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@user170231 Yes. I thought that was obvious, sorry, my bad. – kormoran Apr 05 '21 at 16:49
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No problem, I just had not seen that convention before. – user170231 Apr 05 '21 at 16:50
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@user170231, I have not seen it anywhere as well. :) I just decided to use this symbol. Sometimes I also use a question mark (?), e.g. $40\sqrt2 \text{ ? } 57$ when comparing numbers. It's clear for me, that's why I use it. Hope we cleared this out. – kormoran Apr 05 '21 at 16:51
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3Square this number, you get 100... – Yuval Apr 05 '21 at 16:55
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You have a few general apporaches to similar problem https://math.stackexchange.com/q/816462/130953 might be helpful. – kingW3 Apr 06 '21 at 11:18
5 Answers
$$A=\sqrt{57-40\sqrt2}-\sqrt{40\sqrt2+57} = \sqrt{(4\sqrt2-5)^2} - \sqrt{(4\sqrt2+5)^2} $$
$$ = (4\sqrt2-5) - (4\sqrt2+5) = -10$$
So $A$ is the integer $-10$.
It's just written in some slightly convoluted form.

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Thank you for the response! So we are actually using my comparison and it's necessary for the solution. – kormoran Apr 05 '21 at 17:03
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Kind of. You just need to just show that $4\sqrt{2}-5$ is positive, that's all. – peter.petrov Apr 05 '21 at 17:03
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Finding out what is squared, that's the point. Although your answer has been upvoted, how did you find these numbers? I would like to ask. Yes, we all know that this must be "square of something". But nobody can find that right away. We need a method. This method is used in some places.. – lone student Apr 05 '21 at 17:28
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@lonestudent, they edited the answer. My question is the same. If I knew $57-40\sqrt2=(4\sqrt2-5)^2$, I wouldn't have asked the question in first place. – kormoran Apr 05 '21 at 17:52
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@lonestudent Just some simple 8th grade algebra. You assume it's $(a \pm b\sqrt{2})^2$ and you form a system as Mr. Santos explained. I don't see a big deal here. One just has to play with the algebra/calculations a bit. – peter.petrov Apr 05 '21 at 17:53
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As far as I remember in your first comment you squared $A$ to get $A^2=100$. Now we see the same thing as in Mr Santos' comment. – kormoran Apr 05 '21 at 17:54
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Yes. What I want to say is I think it's the meat of the answer to say where the numbers come from. – lone student Apr 05 '21 at 17:55
That number is $-10$. In fact, if you try to express $\sqrt{57-40\sqrt2}$ as $a+b\sqrt2$, you solve the system$$\left\{\begin{array}{l}a^2+2b^2=57\\2ab=-40.\end{array}\right.$$You will get that $a=5$ and that $b=-4$. By the same method, you will get that $\sqrt{57+40\sqrt2}=-5+4\sqrt2$. So$$\sqrt{57-40\sqrt2}-\sqrt{57+40\sqrt2}=-5+4\sqrt2-\left(5+4\sqrt2\right)=-10.$$

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Thank you for the response! I didn't get why we are trying to express $57-40\sqrt2$ as $a+b\sqrt2$. Isn't it actually in this form when $a=57, b=-40$? – kormoran Apr 05 '21 at 17:01
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I still don't get it. Why would we want to express the roots as $a+b\sqrt2$? (exactly in this particular form) What is the idea behind this? – kormoran Apr 05 '21 at 17:05
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Because then $\sqrt{57+40\sqrt2}=-a+b\sqrt2$, and therefore $\sqrt{57-40\sqrt2}-\sqrt{57+40\sqrt2}=2a$. – José Carlos Santos Apr 05 '21 at 17:07
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It comes from$$57-40\sqrt2=\left(a+b\sqrt2\right)^2=a^2+2b^2+2ab\sqrt2.$$ – José Carlos Santos Apr 05 '21 at 17:10
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Okay, thank you. How do we note that $\sqrt{57-40\sqrt2}$ will then be $\color{red}{-}a+b\sqrt{2}$, though? – kormoran Apr 05 '21 at 17:11
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If $(a+b\sqrt2)^2=57-40\sqrt2$, then $(-a+b\sqrt2)^2=57+40\sqrt2$. So, $\sqrt{57+40\sqrt2}=\pm\left(-a+b\sqrt2\right)$. In this case, the plus sign is the right one. – José Carlos Santos Apr 05 '21 at 17:13
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Thank you for the explanations. I appreciate it. Can we do something else - we can try to write $57-40\sqrt2$ as $(a-b)^2=a^2-2ab+b^2$ (same with the other root). Then we will have (like you did) the system $$\begin{cases}a^2+b^2=57\2ab=40\sqrt2\end{cases}.$$ This is confusing for me. Can't actually $2ab=57$ or something else? – kormoran Apr 05 '21 at 17:16
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First of all is $40\sqrt 2$ greater or less than $57$? One assumes it's a little less.
$(\frac {57}{40})^2 = \frac {(60-3)^2}{1600} = \frac {3600 +9 -2\cdot 3\cdot 60}{1600}= \frac {3600-360+ 9}{1600}= \frac {3249}{1600} > 2$ so $40\sqrt 2 < 57$.
Well if you square $A$ you get
$|40\sqrt 2 + 57| + |40\sqrt 2 - 57|-2\sqrt(|40\sqrt 2+57||40\sqrt 2-57|)=$
$(57+ 40\sqrt 2) + (57 - 40 \sqrt 2) - 2\sqrt{(57+40\sqrt 2)(57-40\sqrt 2)}=$
$57 + 57 - 2\sqrt{57^2-(40\sqrt 2)^2}=$
$114 - 2\sqrt{57^2 - 1600\cdot 2}=$
$114 -2 \sqrt {3249 - 3200} =$
$114 - 2\sqrt{49} = $
$114 - 2\cdot 7= $
$114 - 14= 100$.
So $A^2 = 100$ and $A = \pm 10$.
How cute.
.....
Oh, how clever....$57 \pm 40\sqrt 2 = 57 \pm 2\times 20\sqrt 2= 57 \pm 2\times a(b\sqrt 2)$ where $ab = 20$.
To solve $a^2 + 2b^2 = 57$ we have $57 = 25 + 2*16$ and so $57\pm 2\times 40 \sqrt 2 = 5^2 \pm 2\times 5\cdot 4\sqrt 2 + (4\sqrt 2)^2 = (5 \pm 4\sqrt 2)^2$.
So the who thing works out.
$A =(4\sqrt 2 -5) -(4\sqrt 2 + 5) = -10$.
Cute.

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The check you're doing is indeed necessary in order to remove the absolute value.
However, when you arrive at $3200\mathrel{\Diamond}3249$ you can realize that the difference is a square, which is precisely the condition for a radical of the form $$ \sqrt{a\pm\sqrt{b}} $$ can be “denested”. Let's see why. Suppose $\sqrt{a+\sqrt{b}}=\sqrt{x}+\sqrt{y}$; after squaring we get $$ a+\sqrt{b}=x+y+\sqrt{4xy} $$ and if we equate the two parts we get $$ \begin{cases} x+y=a \\[6px] 4xy=b \end{cases} $$ Hence $(x-y)^2=(x+y)^2-4xy=a^2-b$. So, in order to find the integers $x,y$, we need that $a^2-b$ is a square. Once we have it, we can determine $x$ and $y$.
Note that the same holds for $\sqrt{a-\sqrt{b}}=\sqrt{x}-\sqrt{y}$.
In your case $a=57,b=3200$ and $a^2-b=49=7^2$, so you get $$ x+y=57,\quad x-y=7 $$ and therefore $x=32,y=25$, from which $$ \sqrt{57\pm40\sqrt{2}}=\sqrt{32}\pm5 $$ and you can finish.

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Thank you for the response! I am not really sure I get why we can conclude that the difference is a square when at $3200\Diamond 3249$? The difference between these numbers is $49=7^2$, ok, but we are comparing $40\sqrt2$ and $57$ here. Which numbers have radical form $\sqrt{a\pm\sqrt{b}}$? What does this mean - a number has a radical form "..."? I am not sure how we get the system $$\begin{cases}x+y=a\4xy=b\end{cases}$$ from $$a+\sqrt{b}=x+y+\sqrt{4xy}.$$ Can't $a=y+\sqrt{4xy}$ or something like that? – kormoran Apr 06 '21 at 21:58
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1@Medi It could be something else, but why not looking for the easy solution? – egreg Apr 06 '21 at 22:03
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May I ask you what does a number has radical form $\sqrt{a\pm\sqrt{b}}$ mean? – kormoran Apr 06 '21 at 22:05
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@Medi I'm not sure I understand. Your given numbers have that form, don't they? – egreg Apr 06 '21 at 22:06
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But why squares have this radical form? - "the difference is a square, which is precisely the condition for a radical of the form" – kormoran Apr 06 '21 at 22:07
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1@Medi When you have $\sqrt{a+\sqrt{b}}$ (with integer $a$ and $b$), then the condition for $\sqrt{a+\sqrt{b}}=\sqrt{x}+\sqrt{y}$ with $x,y$ rational, is that $a^2-b$ is a square. I showed that the condition is sufficient, you can try and see it's also necessary. – egreg Apr 06 '21 at 22:11
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Thank you! So we are showing that a radical of the form $\sqrt{a\pm\sqrt{b}}$ can be denested iff $a^2-b$ is a square? – kormoran Apr 06 '21 at 22:53
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May I ask you why after the system you are forming the difference $(x-y)^2=a^2-b$? What is the idea behind this? Why exactly $(x-y)^2$? Can't we still solve it when $a^2-b$ is not a square, e.g. $a^2-b=13$? Why do we want $x,y$ to be integers? – kormoran Apr 06 '21 at 23:06
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@Medi You basically want to find two numbers knowing their sum and their product, $x+y=a,xy=b/4$. This is the same as solving a quadratic equation, whose discriminant is exactly the square of the difference of the roots. – egreg Apr 07 '21 at 07:30
Actually it is not necessary to know if $\;40\sqrt2\;$ is greater or less than $\;57$. Moreover we do not need to square $\;A\;$ or to solve any system of equations.
The check that the original poster does about the comparison between $\;40\sqrt2\;$ and $\;57\;$ is not necessary, indeed just the same we can remove the absolute value inside the first radical as soon as we get the square of a subtraction.
$\begin{align} A&=\sqrt{\left|40\sqrt2-57\right|}-\sqrt{\left|40\sqrt2+57\right|}=\\ &=\sqrt{\left|40\sqrt2-32-25\right|}-\sqrt{40\sqrt2+32+25}=\\ &=\sqrt{\left|40\sqrt2-\left(4\sqrt2\right)^2-5^2\right|}-\sqrt{40\sqrt2+\left(4\sqrt2\right)^2+5^2}=\\ &=\sqrt{\left|\left(4\sqrt2\right)^2+5^2-40\sqrt2\right|}-\sqrt{\left(4\sqrt2\right)^2+5^2+40\sqrt2}=\\ &=\sqrt{\left|\left(4\sqrt2-5\right)^2\right|}-\sqrt{\left(4\sqrt2+5\right)^2}=\\ &=\sqrt{\left(4\sqrt2-5\right)^2}-\sqrt{\left(4\sqrt2+5\right)^2}=\\ &=\left|4\sqrt2-5\right|-\left(4\sqrt2+5\right)=\\ &\underset{\color{blue}{\overbrace{\text{but }\;4\sqrt2-5=\sqrt{32}-\sqrt{25}>0}}}{=}\left(4\sqrt2-5\right)-\left(4\sqrt2+5\right)=\\ &=-10\;. \end{align}$
Hence $A$ is the integer $-10$.

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We are expressing $57=32+25$ in order to obtain the squares of an addition and a subtraction and simplify the two radicals. This method is called “completing the square” and it is used to simplify radicals and also to solve quadratic equations. – Angelo Apr 07 '21 at 11:39