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

Question

(\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 Q1.JPG

Bunuel · Accepted Answer

(\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 . 
___________________________________

(\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.

BrentGMATPrepNow · Answer

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 =  x ² + 2 x  y  +  y ²

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

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

Answer: E

fskilnik · Answer

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

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

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

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

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

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

Regards,
fskilnik.

jaysonbeatty12 · Answer

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

Hoozan · Answer

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

BrentGMATPrepNow · Answer

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)

KarishmaB · Answer

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

= (\sqrt{4 + 5 +4 \sqrt{5}}+\sqrt{4 + 5 -4 \sqrt{5}})^2

= (\sqrt{(2 + \sqrt{5})^2}+\sqrt{(2 - \sqrt{5})^2})^2

= (|2 + \sqrt{5}| +|2 - \sqrt{5}|)^2

= (2 + \sqrt{5} + \sqrt{5} - 2)^2

= (2 \sqrt{5})^2 = 20

Answer (E)

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

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