√(9 + 4√5) + √(30 − 10√5)

Question

\sqrt{9 + 4√5} + \sqrt{30 − 10√5}   =

A) 7

B)  3 + 2\sqrt{5}

C)  19.5 − 3\sqrt{5}

D)  39 − 6\sqrt{5}

E)  140 − 60\sqrt{5}

chetan2u · Accepted Answer

Approximate value
 \sqrt{5}=2.2 
 \sqrt{9 + 4√5} + \sqrt{30 − 10√5}   
 \sqrt{9 + 4*2.2} + \sqrt{30 − 10*2.2}   
 \sqrt{17.8} + \sqrt{7.8}

This is surely greater than 4+2 or 6 and less than 5+3 or 8. 
So A or B. Difficult to choose any one between them visually but probability is 50% now.

Solve the expression

It is a square root so we have to check whether we can make the terms that are under square root as squares.

\sqrt{9 + 4√5} + \sqrt{30 − 10√5}   =x

\sqrt{4+ 5+ 2*2*√5} +\sqrt{25+5 − 2*5√5}

\sqrt{2^2+ √5 ^2+ 2*2*√5} +\sqrt{5^2+ √5^2 − 2*5√5}

\sqrt{(2+ √5)^2}+\sqrt{(5-√5)^2}

2+ √5+5-√5

7

A

Oppenheimer1945 · Answer

\sqrt{(a^2+b^2+2ab)}=|a+b|
\sqrt{(a^2+b^2-2ab)}=|a-b|

4\sqrt{5}=2ab
=>ab=2\sqrt{5}
2^2+(\sqrt{5})^2=9

10\sqrt{5}=2ab
=>5\sqrt{5}=ab
=>5^2+(\sqrt{5})^2=30

T1=2+\sqrt{5}

T2=(5-\sqrt{5})

T1+T2=7

BrushMyQuant · Answer

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

https://www.youtube.com/watch?v=a4xRWnM8g0I

\sqrt{9 + 4√5} + \sqrt{30 − 10√5}   =

Since we have nested roots so idea will be to express the values inside the bigger square root as square of some number

9 + 4√5  => Lets express this as  (a+b)^2 = a^2 + b^2 + 2ab 
 30 − 10√5  => Lets express this as  (a-b)^2 = a^2 + b^2 - 2ab

4√5 = 2ab = 2 * 2 * √5 => a = 2 and b = √5
 10√5 = 2ab = 2 * 5 * √5 => a = 5 and b = √5

=>   \sqrt{9 + 4√5} + \sqrt{30 − 10√5}    =  \sqrt{4 + 5 + 2 * 2 * √5 } + \sqrt{25 + 5 − 2 * 5 * √5 }  
=  \sqrt{2^2 + (√5)^2 + 2 * 2 * √5 } + \sqrt{5^2 + (√5)^2 − 2 * 5 * √5 }  
 =  \sqrt{(2 + √5)^2} + \sqrt{(5 - √5)^2}  
= 2 + √5 + 5 - √5
= 2 + 5 = 7

So,  Answer will be A 
Hope it helps!

Watch the following video to MASTER Roots

https://www.youtube.com/watch?v=5oAec-Y6kOQ

Vivek1707 · Answer

Solved it by approximation. A quick method that can be used in case algebra doesn't click in the time pressure.

√5 is approximately 2.2

So √(9 + 4*2.2) + √(30 - 10*2.2)

√(9+8.8) + (30 - 10*2.2)

√(17.8) + √(8)

√(17.8) ---> ~4.1 or 4.2 i.e. slightly more than 4

√(8) ---> slightly less than 3 i.e. 2.8 or 2.9

Hence √(17.8) + √(8) = ~4.2 + ~2.8 = ~7

Now lets check the other answers
B ----> 3 + 2√5 ---> 3 + 2* 2.2 = 3 + 4.4 i.e. 7.4 --> its very close keep
C ----> 19.5 - 6.6 way more than 7 out
D ----> 39 - 6 * 2.2 = 39 - 13.2 way more than 7 out 
E ----> 140 - 132 close to 8 out

If have to choose between A and B I ll go with A cause the value is very close to 7 and B is definitely more than 7.4. Hence A

√(9 + 4√5) + √(30 − 10√5)

Prep Toolkit

Top 5 GMAT Debrief Videos

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

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

√(9 + 4√5) + √(30 − 10√5)

Prep Toolkit

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

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards