SOLUTION: Given that x and y are integers and {{{sqrt(8sqrt(11)+27) = y-sqrt(x-2y)}}}, find {{{-y^x}}}

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Question 1199848: Given that x and y are integers and sqrt%288sqrt%2811%29%2B27%29+=+y-sqrt%28x-2y%29, find -y%5Ex
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

sqrt%288%2Asqrt%2811%29%2B27%29+=+a%2Bb%2Asqrt%28c%29 where a,b,c are integers and c+%3E=+0

%28sqrt%288%2Asqrt%2811%29%2B27%29%29%5E2+=+%28a%2Bb%2Asqrt%28c%29%29%5E2

8%2Asqrt%2811%29%2B27+=+a%5E2%2B2ab%2Asqrt%28c%29%2Bb%5E2%2Ac

27%2B8%2Asqrt%2811%29+=+%28a%5E2%2Bb%5E2%2Ac%29%2B2ab%2Asqrt%28c%29

Equate corresponding terms to form this system of equations
system%2827+=+a%5E2%2Bb%5E2c%2C8+=+2ab%2C11+=+c%29
Use c = 11 and we get this slightly reduced system
system%2827+=+a%5E2%2B11b%5E2%2C8+=+2ab%29
which turns into
system%2827+=+a%5E2%2B11b%5E2%2C4+=+ab%29
Through a bit of trial and error, or the substitution method, you should find these solutions
a = 4
b = 1

So,
sqrt%288%2Asqrt%2811%29%2B27%29+=+a%2Bb%2Asqrt%28c%29

sqrt%288%2Asqrt%2811%29%2B27%29+=+4%2B1%2Asqrt%2811%29

sqrt%288%2Asqrt%2811%29%2B27%29+=+4%2Bsqrt%2811%29 ... can be verified using WolframAlpha

and
sqrt%288%2Asqrt%2811%29%2B27%29+=+y-sqrt%28x-2y%29

4%2Bsqrt%2811%29+=+y-sqrt%28x-2y%29

At first things seem fine.
The 4 and y match up to show y = 4
But the square root terms do not match up fully.
The left hand side (LHS) has a positive square root, while the right hand side (RHS) has the square root as negative.
This is a contradiction that you'll need to ask your teacher about for further clarification.

If we ignored the positive and negative discrepancy, then we can equate radicands, plug in y = 4 and solve for x.
11 = x-2y
11 = x-2*4
11 = x-8
x = 11-8
x = 19

But as mentioned before, we run into a contradiction
4%2Bsqrt%2811%29+=+y-sqrt%28x-2y%29
4%2Bsqrt%2811%29+=+4-sqrt%2819-2%2A4%29
4%2Bsqrt%2811%29+=+4-sqrt%2811%29
7.316625+=+0.683375

It's possible your teacher made a typo somewhere.
I would ask them for clarification.

The tutor Edwin McCravy reached a similar conclusion in the link below.
https://www.algebra.com/algebra/homework/Radicals/Radicals.faq.question.1151181.html