If a

Question

If  a < b  and  \sqrt{1 + \sqrt{21 + 12\sqrt{3}}} = \sqrt{a} + \sqrt{b} , then what is the value of  a ?

A. 1 
B. 2 
C. 3 
D. 4 
E. 5

Are You Up For the Challenge: 700 Level Questions

GMATWhizTeam · Accepted Answer

Solution:  
 • a < b.
•  \sqrt{1+\sqrt{21+12 \sqrt{3}}} = \sqrt{a} + \sqrt{b}  
Let’s simplify  \sqrt{1 + \sqrt{21 + 2*6\sqrt{3}}}  first.
 •  \sqrt{1 + \sqrt{21 + 2\sqrt{36*3}}} 
•  \sqrt{1 + \sqrt{(\sqrt{12})^2 + (\sqrt{9})^2 + 2\sqrt{12}\sqrt{9}}} 
•  \sqrt{1 + \sqrt{(\sqrt{12} + \sqrt{9})^2}} 
•  \sqrt{1 + \sqrt{12} + \sqrt{9}}  
•  \sqrt{1 + 2\sqrt{3} + 3} 
•  \sqrt{1^2 + 2\sqrt{3} + (\sqrt{3})^2} 
•  \sqrt{(1 + \sqrt{3})^2} 
•  1 + \sqrt{3}  
Now,  (1 + \sqrt{3}) = \sqrt{a} + \sqrt{b} 
After comparing both the sides, we get
 •  a = 1    
Hence, the correct answer is  Option A.

shameekv1989 · Answer

I just want to understand whether there is any possibility of such questions to ever come in GMAT exam? This requires so much of manipulation that it might take more than 5 mins to just understand the manipulation part. And then comes the solution which might require another 5 mins.

Kindly advise.

Kindly advise.

GMATWhizTeam · Answer

Hey Shameekv, I can understand that this might seem like a difficult question. However, if you understand the right methodology to solve this question, it won't take you more than 2-2.5 mins. Also, there is an official question that is based on this kind of question. Here's the link: https://gmatclub.com/forum/the-number-c ... 45521.html If you'll go to this question, you'll see that this question belongs to Official Quant Review 2019. This is proof that you may come across questions like these. Now, as far as solving such questions are concerned, you just need to remember one thing, i.e. you need to use the formula (a +b)^2 = a^2 + b^2 + 2*a*b or (a - b)^2 = a^2 + b^2 - 2*a*b to solve the question. The question is basically trying to test, whether the student can simplify the given expression using the above formula or not. The trick to solve this question is to focus on the square root part first. Notice how I focused on 12 \sqrt{3} first and separated 2 from it and then tried to bring the rest under a square root and manipulate it to get 2\sqrt{12}\sqrt{9} I am not saying that doing all this is easy. But with practice and knowing the above logic, you can definitely solve such question. Also, this is a 700-level question, so even if we take about 3 minutes to solve it, it's okay. As long as we are saving time elsewhere.

sambitspm · Answer

21+12*sqrt(3) = 21+ 2*2*3*sqrt(3)
9+12+ 2*3*(2*sqrt(3))

Expression is Sqrt(1+ 2sqrt(3) + 3)
1+ sqrt(3)
= sqrt(a) + sqrt(b)
as b> a
so a = 1, b = 3

Can also be seen in attachment

dave13 · Answer

If a < b and \sqrt{1 + \sqrt{21 + 12\sqrt{3}}} = \sqrt{a} + \sqrt{b} , then what is the value of a ?

A. 1
B. 2
C. 3
D. 4
E. 5

Are You Up For the Challenge: 700 Level Questions

i solved it in one step

and chose A option. Should some expert confirm if my logic is correct i would appreciate

so just square both sides and one can clearly see without hallucination that a = 1

(\sqrt{1 + \sqrt{21 + 12\sqrt{3}}})^2 = (\sqrt{a} + \sqrt{b})^2

= (1 + \sqrt{21 + 12\sqrt{3}}})= a+2\sqrt{a*b} +b

rvgmat12 · Answer

Breakdown the expression-
root(1+ root((21+12root(3)))) = root(1+ root( (2root(3) + 3)^2))
= 1+root(3)
a = 1, b = 3

BrushMyQuant · Answer

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

Given that \(a < b\) and \(\sqrt{1 + \sqrt{21 + 12\sqrt{3}}} = \sqrt{a} + \sqrt{b}\) and we need to find the value of a

Lets start by simplifying \(\sqrt{1 + \sqrt{21 + 12\sqrt{3}}}\)

To simplify this we can start with the inner most square root, and the idea is to convert the terms inside the square root into sqaure of sum of two numbers like \((x + y)^2\)

\(\sqrt{1 + \sqrt{21 + 12\sqrt{3}}}\) = \(\sqrt{1 + \sqrt{9 + 12 + 2 * 3 * 2\sqrt{3}}}\) = \(\sqrt{1 + \sqrt{3^2 + (2\sqrt{3})^2 + 2 * 3 * 2\sqrt{3}}}\)
= \(\sqrt{1 + \sqrt{(3 + 2\sqrt{3})^2}}\) = \(\sqrt{1 + 3 + 2\sqrt{3}}\) = \(\sqrt{4 + 2\sqrt{3}}\)

Following similar logic lets try to write \(4 + 2\sqrt{3}\) like \((x + y)^2\)

=> \(\sqrt{4 + 2\sqrt{3}}\) = \(\sqrt{1 + 3 + 2*1*\sqrt{3}}\) = \(\sqrt{1^2 + (\sqrt{3})^2 + 2*1*\sqrt{3}}\) = \(\sqrt{(1 + (\sqrt{3})^2}\)
= \(1 +\sqrt{3}\) = \(\sqrt{a} + \sqrt{b}\)

Now, a < b
=> \(\sqrt{a}\) = 1 ( as 1 < \(\sqrt{3}\))
=> a = 1

So, Answer will be A
Hope it helps!

Watch the following video to MASTER Roots

bumpbot · Answer

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