What is the value of |3^(1/2) - |5^(1/2) - 7^(1/2)|| - ||3^(1/2)

Question

What is the value of  |\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|  ?

A.  2\sqrt{5}-2\sqrt{7}-2\sqrt{3}

B.  2\sqrt{5}-2\sqrt{7}

C.  2\sqrt{7}-2\sqrt{5}

D.  2\sqrt{5}-2\sqrt{7}+2\sqrt{3}

E.  2\sqrt{7}+2\sqrt{5}

Bunuel · Accepted Answer

Official Solution:

To answer this question we should recall the property of the absolute value:

|x| = x , when  x \geq 0 ;
 |x| = -x , when  x < 0 .

So, we should evaluate the expressions in the modulus to see whether they are positive or negative.

STEP 1:

Since  \sqrt{5}-\sqrt{7}<0 , then  |\sqrt{5}-\sqrt{7}|=-(\sqrt{5}-\sqrt{7})=\sqrt{7}-\sqrt{5} ;
Since  \sqrt{3} - \sqrt{5}<0 , then  |\sqrt{3} - \sqrt{5}|=-(\sqrt{3} - \sqrt{5})= \sqrt{5} -\sqrt{3}  .

Thus,  |\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|  will become:

|\sqrt{3} -(\sqrt{7}-\sqrt{5})| - |(\sqrt{5} -\sqrt{3})-\sqrt{7}|=|\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}| .

STEP 2:

\sqrt{3} +\sqrt{5}-\sqrt{7}  must be positive because  \sqrt{3} +\sqrt{5}=1.something + 2.something=3.something , while  \sqrt{7}<3.something . So,  |\sqrt{3} +\sqrt{5}-\sqrt{7}| =\sqrt{3} +\sqrt{5}-\sqrt{7} ;

\sqrt{5} -\sqrt{3}-\sqrt{7}  is obviously negative. So,  \sqrt{5} -\sqrt{3}-\sqrt{7}=-(\sqrt{5} -\sqrt{3}-\sqrt{7})=\sqrt{7}+\sqrt{3}-\sqrt{5}

Thus,  |\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}|  will become:

(\sqrt{3} +\sqrt{5}-\sqrt{7})-(\sqrt{7}+\sqrt{3}-\sqrt{5})=2\sqrt{5}-2\sqrt{7} .

Answer: B.

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem ↧↧↧

We need to find the value of \(|\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|\)

This problem is based on the theory
|x| = x if x ≥ 0
|x| = -x if x ≤ 0

\(|\sqrt{5}-\sqrt{7}|\) = -(\(\sqrt{5}-\sqrt{7}\)) as \(\sqrt{5} < \sqrt{7}\)
\(|\sqrt{3}-\sqrt{5}|\) = -(\(\sqrt{3}-\sqrt{5}\)) as \(\sqrt{3} < \sqrt{5}\)

=> \(|\sqrt{3} - (-(\sqrt{5}-\sqrt{7})| - |\sqrt{5} - \sqrt{3}-\sqrt{7}|\)
= \(|\sqrt{3} + \sqrt{5}-\sqrt{7}| - |\sqrt{5} - \sqrt{3}-\sqrt{7}|\)
( As \(\sqrt{3} + \sqrt{5} - \sqrt{7} > 0 \) and \(\sqrt{5} - \sqrt{3}-\sqrt{7}\) < 0 )

= \(\sqrt{3} + \sqrt{5}-\sqrt{7} - (-(\sqrt{5} - \sqrt{3}-\sqrt{7}))\) )
= \(\sqrt{3} + \sqrt{5}-\sqrt{7} + \sqrt{5} - \sqrt{3}-\sqrt{7}\)
= \(2\sqrt{5} - 2\sqrt{7}\)

So, Answer will be B
Hope it helps!

Playlist on Solved Problems on Absolute Values here

Watch the following video to MASTER Absolute Values

https://www.youtube.com/playlist?list=PLj7wQVQKu1e0RkZMS9RJ_2D_kqPuKiioD

gmatophobia · Answer

Concept Tested

|x| = x ; x is non-negative

|x| = -x ; x is negative

Question

|\sqrt 3 - |\sqrt 5 - \sqrt 7||- ||\sqrt 3 - \sqrt 5|- \sqrt 7|

Solution

\sqrt 5 - \sqrt 7  is negative as  \sqrt 5 < \sqrt 7

\sqrt 3 - \sqrt 5  is negative as  \sqrt 3 < \sqrt 5

So, when we open the mod, the value will be - ( \sqrt 5 - \sqrt 7 ) i.e.  \sqrt 7 - \sqrt 5

Similarly for  \sqrt 3 - \sqrt 5 , the resultant value will be  \sqrt 5 - \sqrt 3

|\sqrt 3 - (\sqrt 7 - \sqrt 5)|- |(\sqrt 5 - \sqrt 3)- \sqrt 7|

|\sqrt 3 - \sqrt 7 + \sqrt 5|- |\sqrt 5 - \sqrt 3 - \sqrt 7|

The first term is positive and the second term is negative, hence once we open the modulus the resultant expression becomes

(\sqrt 3 - \sqrt 7 + \sqrt 5) - (-\sqrt 5 + \sqrt 3 + \sqrt 7))

\sqrt 3 - \sqrt 7 + \sqrt 5 + \sqrt 5 - \sqrt 3 - \sqrt 7

- 2\sqrt 7 + 2\sqrt 5

=  2\sqrt 5 - 2\sqrt 7

IMO Option B

Barfi · Answer

When a term is negative, why are the signs of its values inside the bracket swapped while opening the modulus?

For instance:

3√−7√+5√|−|5√−3√−7√||3−7+5|−|5−3−7|

The first term is positive and the second term is negative, hence once we open the modulus the resultant expression becomes

(3√−7√+5√)−(−5√+3√+7√))(3−7+5)−(−5+3+7))

chiplesschap · Answer

Did it in a crude way.
 sqrt (3) = 1.73, sqrt (5) ~ 2.2, sqrt (7) : ~2.6

first expression becomes : | 1.73 - |-0.4|| = |1.73 - 0.4| = 1.33

second expression becomes : ||1.73 - 2.2|| - 2.6| = ||-0.5|-2.6|| = |0.5-2.6| = |-2.1| = 2.1

therefore result should be a negative value i.e. ~-0.78

Only A & B work. B is closest -> (2*2.2-2*2.6) = ~-0.8

Piqueee · Answer

This question actually came on the official GMAT for me. I did the 'crude' way shown above and I got it right. During the exam, you are limited on time so got to find creative ways to solve. It worked out lol!

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What is the value of |3^(1/2) - |5^(1/2) - 7^(1/2)|| - ||3^(1/2)

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What is the value of |3^(1/2) - |5^(1/2) - 7^(1/2)|| - ||3^(1/2)

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My Rewards