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!

BlackHat098 · Answer

hello, do you have links to more questions like this, which is heavy on roots.

Bunuel · Answer

Exponents and Roots of Numbers

Theory: 
 
  Exponents and Roots on the GMAT: Tips and hints  
  Cyclicity and the remainders on the GMAT

Questions 
 
  Units digits, exponents, remainders problems  
  Tough and tricky DS exponents and roots questions with detailed solutions  
  Tough and tricky PS exponents and roots questions with detailed solutions  
  Exponents PS Questions  
  Roots PS Questions

Check below for more: 
 
  ALL YOU NEED FOR QUANT ! ! !  
  Ultimate GMAT Quantitative Megathread

Hope this helps.

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)

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards