M25-36

Question

Which of the following is equal to  \frac{1}{2 - \sqrt{3}} ?

A.  \sqrt{3} - 2  
B.  2 + \sqrt{3}  
C.  \sqrt{2} + \sqrt{3}  
D.  2 - \sqrt{3}  
E.  \sqrt{3} + 4

Bunuel · Answer

Official Solution:

Which of the following is equal to  \frac{1}{2 - \sqrt{3}} ?

A.  \sqrt{3} - 2  
B.  2 + \sqrt{3}  
C.  \sqrt{2} + \sqrt{3}  
D.  2 - \sqrt{3}  
E.  \sqrt{3} + 4

Rationalize the fraction by multiplying both the numerator and the denominator by  (2 + \sqrt{3}) . The purpose of rationalization is to eliminate irrational expressions in the denominator and simplify the expression:
 
 \frac{2 + \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} =  
 
 =\frac{2 + \sqrt{3}}{4 - 3} =  
 
 =2 + \sqrt{3} .

Answer: B

bsmith37 · Answer

Bunuel,

I'm just curious about what the intuition is behind this type of solution. How do we know we should multiply by  (2 + \sqrt{3})  as opposed to some other expression, or rather than just solving every answer in raw form?

Bunuel · Answer

This algebraic manipulation is called rationalisation and is performed to eliminate irrational expression in the denominator. For this particular case we are doing rationalisation by applying the following rule: (a-b)(a+b)=a^2-b^2 . The concept of rationalisation of a fraction used to solve this problem is often tested on the GMAT. Questions relied on this technique: if-x-0-then-106291.html if-n-is-positive-which-of-the-following-is-equal-to-31236.html consider-a-quarter-of-a-circle-of-radius-16-let-r-be-the-131083.html in-the-diagram-not-drawn-to-scale-sector-pq-is-a-quarter-139282.html in-the-diagram-what-is-the-value-of-x-129962.html the-perimeter-of-a-right-isoscles-triangle-is-127049.html which-of-the-following-is-equal-to-98531.html if-x-is-positive-then-1-root-x-1-root-x-163491.html Hope it helps.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

M25-36

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