What is the value of (10/(3*11^(1/2) + 10) + 30*11^(1/2))^(1/2)

Question

What is the value of  \sqrt{\frac{10}{3\sqrt{11} + 10} + 30\sqrt{11}}

A.  \sqrt{11}

B.  \sqrt{33}

C. 8

D. 10

E.  30\sqrt{11}

This is a  PS Butler Question

Check the links to other Butler Projects: 
 
   Data Sufficiency Butler 
   Critical Reasoning Butler  
   Sentence Correction Butler  
   Reading Comprehension Butler  
   Integrated Reasoning Butler

wickedvikram · Answer

D 
It's a direct solve, take lcm of the denominator. Even rationalising is not needed 10+3sqrt of 11 will cancel and sqrt of 100 will remain

KarishmaB · Answer

\sqrt{\frac{10}{3\sqrt{11} + 10} + 30\sqrt{11}}

Rationalise the denominator first of all.

\sqrt{\frac{10*(10 -3\sqrt{11}) }{(10+3\sqrt{11})*(10 -3\sqrt{11})} + 30\sqrt{11}}

\sqrt{\frac{100 -30\sqrt{11} }{10^2 - 3^2*11} + 30\sqrt{11}}

\sqrt{100 -30\sqrt{11} + 30\sqrt{11}}

\sqrt{100}

= 10

Answer (D)

ThatDudeKnows · Answer

I truly don't understand why anyone would do the "real" math on this question rather than ballparking. Just ballpark. The square root of 11 is somewhere between 3 and 4...closer to 3...let's just try 3.3 and see what happens.

\sqrt{\frac{10}{10 + 10} + 30*3.3}

\sqrt{0.5+100} = 10

(Ahhh, notice that the second term will be the driving influence and it's going to be right around 100...it'll definitely be bigger than 64, and it'll definitely be smaller than 10000...only one answer choice is anywhere close.)

Answer choice D.

There seems to be some sort of pride for most students about doing the "real" math. But there are no bonus points for doing that. Set aside your pride and your need to follow the rules what academic math teachers have taught you...and just get the right answer as quickly as possible without all the steps that give you opportunities to get confused or make a silly mistake.

ThatDudeKnowsBallparking

Beko · Answer

Hello Karishma,

why do you change the order of "3root(11)+10" into "10+3root(11)" in the denominator when rationalizing?

Thank you in advance!

Best regards

BrushMyQuant · Answer

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

https://www.youtube.com/watch?v=rHBDAb5TAqk

\sqrt{\frac{10}{3\sqrt{11} + 10} + 30\sqrt{11}}

Lets solve the first part of the term inside the square root

\frac{10}{3\sqrt{11} + 10}

To solve this we will rationalize the denominator by multiplying the numerator and the denominator with the conjugate of the denominator

=>  \frac{10}{3\sqrt{11} + 10}  =  \frac{10}{3\sqrt{11} + 10} * \frac{3\sqrt{11} - 10}{3\sqrt{11} - 10}

Now, in the denominator we have the terms in the form  (a + b ) * (a - b) = a^2 - b^2

=>  \frac{10 * 3\sqrt{11} - 10 }{(3\sqrt{11})^2 - 10^2} 
=  \frac{30\sqrt{11} - 100 }{9 * 11 - 100} 
 =  \frac{30\sqrt{11} - 100 }{-1} 
 =  100 - 30\sqrt{11}

=>   \sqrt{\frac{10}{3\sqrt{11} + 10} + 30\sqrt{11}}  =   \sqrt{100 - 30\sqrt{11} + 30\sqrt{11}} 
=  \sqrt{100}  = 10

So,  Answer will be D 
Hope it helps!

Watch the following video to learn How to Rationalize Roots

https://www.youtube.com/watch?v=LhR_gXt8CLw

BrushMyQuant · Answer

This was done because  10^2  is greater than  (3\sqrt{11})^2  and we will get -1 in the denominator otherwise.
We can solve it without changing the order also and then multiply with -1 as i did in my solution above.
Hope it helps!

What is the value of (10/(311^(1/2) + 10) + 3011^(1/2))^(1/2)

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