(root(9 + root(80)) + root(9 - root(80)))^2 =

\((\sqrt{9+\sqrt{80}}+\sqrt{9-\sqrt{80}})^2=\)


A. 1

B. \(9 - 4*\sqrt{5}\)

C. \(18 - 4*\sqrt{5}\)

D. 18

E. 20


You should know two properties:

1. \((x+y)^2=x^2+2xy+y^2\), (\((x-y)^2=x^2-2xy+y^2\));

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

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\((\sqrt{9+\sqrt{80}}+\sqrt{9-\sqrt{80}})^2=\)

\(=(\sqrt{9+\sqrt{80}})^2+2(\sqrt{9+\sqrt{80}})(\sqrt{9-\sqrt{80}})+(\sqrt{9-\sqrt{80}})^2=\)

\(=9+\sqrt{80}+2\sqrt{(9+\sqrt{80})(9-\sqrt{80})}+9-\sqrt{80}=\)

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

\(=18+2\sqrt{81-80}=\)

\(=18+2=20\).

Answer: E.

Hope it helps.
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Notice that the expression is in the form (x + y)², where x = √(9 + √80) and y = √(9 - √80)

We know that (x + y)² = x² + 2xy + y²

If x = √(9 + √80), then x² = 9 + √80
If y = √(9 - √80), then y² = 9 - √80
Finally, xy = [√(9 + √80)][√(9 - √80)] = 81 - 80 = 1

So, we get:
(x + y)² = x² + 2xy + y²
= (9 + √80) + 2(1) + (9 - √80)
= 9 + √80 + 2 + 9 - √80
= 20

Answer: E
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\(A = \sqrt {9 + \sqrt {80} }\)
\(B = \sqrt {9 - \sqrt {80} }\)

\(?\,\,\,:\,\,\,{\text{expression}}\,\,{\text{ = }}\,\,{\left( {A + B} \right)^{\text{2}}}{\text{ = }}{{\text{A}}^{\text{2}}} + 2AB + {B^2}\)

\({A^2} = {\left( {\sqrt {9 + \sqrt {80} } } \right)^2} = 9 + \sqrt {80}\)
\({B^2} = {\left( {\sqrt {9 - \sqrt {80} } } \right)^2} = 9 - \sqrt {80}\)

\(AB = \sqrt {9 + \sqrt {80} } \cdot \sqrt {9 - \sqrt {80} } = \sqrt {\left( {9 + \sqrt {80} } \right)\left( {9 - \sqrt {80} } \right)} \,\,\,\mathop = \limits^{\left( * \right)} \,\,\,\sqrt 1 = \boxed1\)
\(\left( * \right)\,\,\,\,\,\left( {9 + \sqrt {80} } \right)\left( {9 - \sqrt {80} } \right)\,\,\, = \,\,\,\,{9^2} - {\left( {\sqrt {80} } \right)^2} = 81 - 80 = 1\)


\(? = \left( {9 + \sqrt {80} } \right) + 2 \cdot \boxed1 + \left( {9 - \sqrt {80} } \right) = 20\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.
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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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Can we solve this question the way we did for this one: https://gmatclub.com/forum/if-m-4-1-2-4 ... l#p1611841

That kind of estimation will help us eliminate two or three answer choices, but I think it would be pretty tricky to use straightforward rationale to justify why answer choice E (20) is a better estimate than answer choice D (18)
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