What is the value of [m][fraction]1/2[/fraction][square_root]√21+12√3[

Question

What is the value of  \frac{1}{2}\sqrt{21+12√3}-√3  ?

(A)  \frac{1}{2}

(B)  \frac{2}{3}

(C)  \frac{3}{2}

(D)  \frac{6}{7}

(E)  \frac{7}{4}

nick1816 · Accepted Answer

\frac{1}{2}\sqrt{21+12√3}-√3

\frac{1}{2}\sqrt{12+9+2√(12*9)}-√3

\frac{1}{2}\sqrt{(√12+√9)^2} - √3

\frac{1}{2}   -√3

= \frac{3}{2}

GMATinsight · Answer

Fastest Method    : APPROXIMATION

\frac{1}{2}\sqrt{21+12√3}-√3=\frac{1}{2}\sqrt{21+12*1.7}-√3 ≈ \frac{1}{2}\sqrt{41}-1.7 = \frac{1}{2}*6.4-1.7 = 3.2-1.7 = 1.5

Answer: Option C

danbrown9445 · Answer

Easy way:
 =\frac{1}{2}\sqrt{21+12√3}-√3 
 =\frac{1}{2}\sqrt{9+12√3+12}-√3 
 =\frac{1}{2}\sqrt{(3)^2+2*3*2√3+(2√3)^2}-√3 
 =\frac{1}{2} \sqrt{(3+2√3)^2}-√3 
 =\frac{1}{2} (3+2√3)-√3 
 =\frac{(3+2√3-2√3)}{2} 
 =\frac{3}{2}

BrushMyQuant · Answer

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

https://www.youtube.com/watch?v=lZBb3mR45bI

We need to simplify  \frac{1}{2}\sqrt{21+12√3}-√3

In order to solve these kind of problems involving nested square roots, we need to simplify the terms inside the bigger square root to convert them to a perfect square

=> We need to write 21+12√3 as  (a+b)^2

Now, we will try to write 21 as  a^2 + b^2  and 12√3 as 2*a*b
=> 12√3 = 2 * 2√3 * 3 
And 21 =  (2√3)^2 + 3^2  = 4*3 + 9 = 12 + 9

=>  21 + 12√3  =  (2√3 + 3)^2 
=>  \frac{1}{2}\sqrt{21+12√3}-√3  =  \frac{1}{2}\sqrt{(2√3 + 3)^2}-√3 
=  \frac{1}{2}(2√3 + 3) - √3  = √3 + 3/2 - √3 = 3/2

So,  Answer will be C 
Hope it helps!

Watch the following video to MASTER Roots

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

luisdicampo · Answer

Deconstructing the Question

We need to evaluate

\frac12\sqrt{21+12\sqrt3}-\sqrt3

Expressions of the form

\sqrt{a+b\sqrt{c}}

can often be written as

\sqrt{m}+\sqrt{n}

because

(\sqrt{m}+\sqrt{n})^2=m+n+2\sqrt{mn}

Step-by-step

Match terms with

\sqrt{21+12\sqrt3}

So

m+n=21

and

2\sqrt{mn}=12\sqrt3

Divide by  2

\sqrt{mn}=6\sqrt3

Square both sides

mn=108

Now solve

t^2-21t+108=0

(t-9)(t-12)=0

So

m=9

n=12

Thus

\sqrt{21+12\sqrt3}=\sqrt9+\sqrt{12}

=3+2\sqrt3

Substitute back

\frac12(3+2\sqrt3)-\sqrt3

=\frac32+\sqrt3-\sqrt3

=\frac32

Answer C: 3/2

What is the value of [m][fraction]1/2[/fraction][square_root]√21+12√3[

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

What is the value of [m][fraction]1/2[/fraction][square_root]√21+12√3[

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