6^(1/2)/(1/2^(1/2)+1/3^(1/2))

Question

\frac{\sqrt{6}}{(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}})}=

A.  \frac{6}{5}

B.  \frac{5}{6}

C. 5

D.  2\sqrt{3} +3\sqrt{2}

E.  6\sqrt{3}-6\sqrt{2}

AaronPond · Accepted Answer

The trick behind this question is to use the answer choices to guide your mathematics. (In other words, don't do math just because you can... do math because it gets you closer to your target!)

You should notice immediately that the answer choices are much simpler. Three of the five answer choices contain no fractions at all. None of the answer choices contain  \sqrt{6} . Use this to your advantage to help you think about the question. One quick way to get rid of both the  \sqrt{6}  and the "fractions inside of fractions" issue is to do what I call "  Multiply by 1  ". In this case, multiply the original expression by  \sqrt{6}/\sqrt{6} . Since  \sqrt{6}/\sqrt{6}=1 , we don't actually change the value; we just change the shape of the expression. Thus,

\frac{\sqrt{6}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}}*(\frac{\sqrt{6}}{\sqrt{6}})=\frac{6}{\sqrt{3}+\sqrt{2}}

Now the expression that remains is starting to look more like the answer choices, but we still need to get rid of the compound denominator  (\sqrt{3}+\sqrt{2}) . Once again, look to the answer choices for clues. Both answer choices  A  and  E  contain a  6 . But it should be clear that answer choice  A  cannot be an option.  (\sqrt{3}+\sqrt{2})  does not equal 5. So, let's try to turn what we have into answer choice  E .

Answer choice E contains a factor of  (\sqrt{3}-\sqrt{2}) , which should immediately get us thinking about the possibility of difference of squares. Multiplying our reduced expression by "1" again gives us:

(\frac{6}{\sqrt{3}+\sqrt{2}})*(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}})

Using the concept of difference of squares, we can quickly see that the denominator of this fraction simplifies down completely. Watch this "disappearing math" trick:

(\sqrt{3}+\sqrt{2})*(\sqrt{3}-\sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3-2 = 1

What began as a "Mathugly" arithmetic issue quickly simplifies down to  6\sqrt{3}-6\sqrt{2} .

The answer is E.

Addendum : As with many quantitative questions on the GMAT, there is "more than one way to skin a CAT" for this question (pun intended.) We can also use the answer choices not just as targets that guide our math, but as leverage for approximating. Watch this. If we recognize that  \sqrt{3}\approx{1.7}  and  \sqrt{2}\approx{1.4} , then the we can approximate our answer very quickly:
 \frac{6}{\sqrt{3}+\sqrt{2}}\approx{\frac{6}{1.4+1.7}}\approx{\frac{6}{3.1}}\approx{2}

Now, we look at the answer choices to see which one is roughly equal to 2.

(A)  \frac{6}{5}    <-- quickly eliminated. Too small.

(B)  \frac{5}{6}    <-- quickly eliminated. Too small.

(C)  5    <-- quickly eliminated. Too big.

(D)  2\sqrt{3} + 3\sqrt{2}    <-- quickly eliminated. Too big.

(E)  6\sqrt{3} - 6\sqrt{2}    <-- The right answer.  6(\sqrt{3} - \sqrt{2})\approx{6(1.7-1.4)}\approx{2}

Anyway you look at it, the answer is still E.

Bunuel · Answer

\frac{\sqrt{6}}{(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}})}=\frac{\sqrt{6}}{(\frac{\sqrt{3}+\sqrt{2}}{\sqrt{2}\sqrt{3}})}=\frac{\sqrt{6}}{(\frac{\sqrt{3}+\sqrt{2}}{\sqrt{6}})}=\frac{\sqrt{6}*\sqrt{6}}{\sqrt{3}+\sqrt{2}}=\frac{6}{\sqrt{3}+\sqrt{2}} .

Multiply by  \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} :

\frac{6}{\sqrt{3}+\sqrt{2}}*\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}=\frac{6*(\sqrt{3}-\sqrt{2})}{3-2}=6*\sqrt{3}-6\sqrt{2} .

Answer: E.

BrentGMATPrepNow · Answer

Nice question!

Here's another approach:

Let's first deal with the denominator: 1/√2 + 1/√3 = √3/√6 + √2/√6 
= (√3 + √2)/√6

So.... \frac{\sqrt{6}}{(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}})}=  √6/

= (√6)(√6)/(√3 + √2)

= 6/(√3 + √2)
 
Now we can use the fact that √3 ≈ 1.7 and √2 ≈ 1.4 to get....
≈ 6/(1.7 + 1.4)
≈ 6/(3.1)

Since 6/3 = 2, we know that 6/(3.1) =  a number a bit smaller than 2

Now check the answer choices to see which one evaluates to be  a number a bit smaller than 2

A.  \frac{6}{5}  ELIMINATE

B.  \frac{5}{6}  ELIMINATE

C. 5 ELIMINATE

D.  2\sqrt{3} +3\sqrt{2}  ≈ 2(1.7) + 3(1.4) ≈ a number that's bigger than 6 ELIMINATE

E.  6\sqrt{3}-6\sqrt{2}   = 6(√3 - √2) ≈ 6(1.7 - 1.4) ≈ 6(0.3) ≈ 1.8 PERFECT!

Answer: E

Cheers,
Brent

KanishkM · Answer

\frac{\sqrt{6}}{(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}})}=

So if you Multiply the ratio with  \sqrt{3}-\sqrt{2}

You will get the ratio as 6 ( \sqrt{3}-\sqrt{2} ) /  \sqrt{3}-\sqrt{2}  *  \sqrt{3}+\sqrt{2}

6 * ( \sqrt{3}-\sqrt{2}  ) / 3-2

Using  a^2 - b^2 = (a-b) (a+b)

6 * ( \sqrt{3}-\sqrt{2} )

E

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6^(1/2)/(1/2^(1/2)+1/3^(1/2))

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