Hi and welcome to Gmat Club. Below is properly formated solution for this problem:

You should know two properties:
1. \((x+y)^2=x^2+2xy+y^2\), (\((x-y)^2=x^2-2xy+y^2\));

2. \((x+y)(x-y)=x^2-y^2\).

\((\sqrt{9+sqrt{80}}+\sqrt{9-\sqrt{80}})^2=(\sqrt{9+\sqrt{80}})^2+2(\sqrt{9+\sqrt{80}})(\sqrt{9-sqrt{80}})+(\sqrt{9-\sqrt{80}})^2=\)
\(=9+\sqrt{80}+2\sqrt{(9+\sqrt{80})(9-\sqrt{80})}+9-\sqrt{80}=18+2\sqrt{9^2-(\sqrt{80})^2}=18+2\sqrt{81-80}=18+2=20\).

Answer: E.

Hope it helps.
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9 + sqrt(80) + 9 -sqrt(80) + 2 sqrt (81-80)

=20

These are the concept that you need to know to solve this problem

(x+y)^2 = x^2 +2xy+y^2
sqrtX times sqrt Y = sqrt XY
(a-b)(a+b) = a^2-b^2

squaring the term gives

9 + sqrt(80) + 9 - sqrt(80 + 2*(81 - 80) = 20

\((sqrt{a} + sqrt{b})^2\) = a + b + \(2*sqrt{ab}\)

Here,
a = \(9 + sqrt{81}\)
b = \(9 - sqrt{81}\)

therefore,
answer = \(9 + [strike]sqrt{81}[/strike]\) + \(9 - [strike]sqrt{81}[/strike]\) + \(2*(81-80)\)
= 9 + 9 + 2 = 20

im still not seeing the 81 and 80 here.....
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"Ignorance more frequently begets confidence than does knowledge" Darwin

If you are pressed for time on this question you can get down to D or E quickly by estimating. Convert the radicals to actual numbers and execute the arithmetic.

The answer choices are approximately
A 1
B ~0
C ~9
D 18
E 20

The question can be restated as

(sqrt(9+9) + sqrt (9-9))squared
sqrt(18)squared
18ish
D or E

Depending on your ability to execute the exponent algebra, it may be a "win" to get this quickly to a coin flip and move on.
A similar approach is useful on OG2019 77 (estimating the value of sqrt 2 and defining the longest and the shortest the cord could be), 81, 100, 119, and 180. 100 and 180 also need a variables in the answer choices approach.

Jayson Beatty
Indigo Prep
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Jayson Beatty
Indigo Prep
http://www.indigoprep.com