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

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Joined: 05 Aug 2008
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(root(9 + root(80)) + root(9 - root(80)))^2 =  [#permalink]

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Updated on: 03 Apr 2019, 07:02
5
33
00:00

Difficulty:

45% (medium)

Question Stats:

62% (01:35) correct 38% (02:06) wrong based on 618 sessions

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

A. 1

B. $$9 - 4*\sqrt{5}$$

C. $$18 - 4*\sqrt{5}$$

D. 18

E. 20

Attachment:

Q1.JPG [ 10.17 KiB | Viewed 11540 times ]

Originally posted by smarinov on 02 Jan 2009, 17:42.
Last edited by Bunuel on 03 Apr 2019, 07:02, edited 3 times in total.
Renamed the topic, edited the question and added the OA.
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Re: (root(9 + root(80)) + root(9 - root(80)))^2 =  [#permalink]

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09 Jun 2010, 13:38
10
8
You should know two properties:
1. $$(x+y)^2=x^2+2xy+y^2$$, ($$(x-y)^2=x^2-2xy+y^2$$);

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

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

Hope it helps.
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Re: (root(9 + root(80)) + root(9 - root(80)))^2 =  [#permalink]

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20 Jul 2017, 07:32
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Top Contributor
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blakemancillas wrote:
$$(\sqrt{9+\sqrt{80}}+\sqrt{9-\sqrt{80}})^2=$$

A. 1
B. 9 - 4*5^1/2
C. 18 - 4*5^1/2
D. 18
E. 20

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

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Re: (root(9 + root(80)) + root(9 - root(80)))^2 =  [#permalink]

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05 Sep 2018, 08:19
1
blakemancillas wrote:
$$(\sqrt{9+\sqrt{80}}+\sqrt{9-\sqrt{80}})^2=$$

A. 1
B. 9 - 4*5^1/2
C. 18 - 4*5^1/2
D. 18
E. 20

$$A = \sqrt {9 + \sqrt {80} }$$
$$B = \sqrt {9 - \sqrt {80} }$$

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

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

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

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

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

Regards,
fskilnik.
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Re: (root(9 + root(80)) + root(9 - root(80)))^2 =  [#permalink]

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05 Sep 2018, 09:41
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.

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
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Re: (root(9 + root(80)) + root(9 - root(80)))^2 =  [#permalink]

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20 Oct 2019, 10:42
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Re: (root(9 + root(80)) + root(9 - root(80)))^2 =   [#permalink] 20 Oct 2019, 10:42
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