Which of the following quantities is the largest?

Question

(A)  \sqrt{2}

(B)  \sqrt {3}

(C)  \sqrt {4}

(D)  \sqrt {5}

(E)  \sqrt {6}

KarishmaB · Accepted Answer

So I am assuming that the choices are:
 2^{\frac{1}{2}}, 3^{\frac{1}{3}}, 4^{\frac{1}{4}}, 5^{\frac{1}{5}}, 6^{\frac{1}{6}}

Since fractional powers are a pain, let me multiply all the powers by 60 (LCM) to make them manageable.

2^{30}, 3^{20}, 4^{15} ( = 2^{30}), 5^{12}, 6^{10} 
Bases and powers, both are different. To compare, we need to make one of them the same.

2^{30} = 8^{10} 
 3^{20} = 9^{10} 
 6^{10}

Obviously, out of these three,  9^{10}  is greatest.

Now we just need to compare  3^{20}  with  5^{12} 
 3^{20} = 243^{4} 
 5^{12} = 125^{4}

3^{20} ( = 3^{\frac{1}{3}})  is the greatest.

jlgdr · Answer

The question shows another answer choice for C, but I second your approach Karishma Now, let's assume the question as is 2^30, 3^20 and 5^15 In this case, I assume it would be a matter of setting all exponents to GCF and compare the bases So we would end up with 64^5, 81^5, 64^5 Clearly now B is the winner here Kudos rain! Cheers! J

Bunuel · Answer

Edited option C. It should be as written in Karishma's post.

KarishmaB · Answer

2^30, 3^20 and 5^15
Yes, you can get the common power of 5

64^5, 81^5 and 125^5

So 5^15 is greatest here.

happyface101 · Answer

I Karishma - it's very clear and simple until the end. To compare 3^20 with 5^12, it seems that we need to know what 3^16 and 5^8 are. Isn't 3^16 a uncommon number to memorize? Or is this expected for the gmat / there is an easier way to compare? Thanks so much!

ENGRTOMBA2018 · Answer

It is not  3^{16}  that you need to remember.

When you end up comparing  3^{20}  and  5^{12} , try to raise the 2 numbers to the same power.

In this case the common GCD of 20 and 12 is 4.

Thus  3^{20}=(3^5)^4  and

5^{12} = (5^3)^4

Giving you the 2 values as  243^4  and  125^4 . So now can easily see that the 2 numbers are raised to the same power but with a differnet base giving you  3^{20} > 5^{12}

Kritisood · Answer

first multiply all the powers by 60 cus fractions are confusing
 2^{30}, 3^{20}, 4^{15}, 5^{12}, 6^{10}

Now, the easiest way as per me to do this would be to start with two powers; compare them and proceed ahead. 
quicky browsing through the ans choices I can estimate the answer must be either A or B so let's start with them.

we need to get the bases or the powers equal to be able to equate them; lets make the powers equal
 2^{30}  can be written as  (2^{3})^{10}  and  3^{20}  can be written as  (3^{2})^{10} 
 2^3<3^2  hence B is larger

now if you want to test further you could, it'll be easier now to compare the remaining three powers with 3^20 since we have already established that it's higher than one answer choice (A)

3^{20} =(3^{4})^{5} 
 4^{15} = (4^{3})^{5} 
again  3^4> 4^3

we can check the rest of the cases the same way.

-- adding a shorter way --

once we get here: 
 2^{30}, 3^{20}, 4^{15}, 5^{12}, 6^{10}

we can take the power of 5 common from all except 5^12

ie.  (2^6)^5, (3^4)^5, (4^3)^5, (6^2)^5 
now we just have to compare 2^6 with 3^4 and similarly for rest

Capablanca · Answer

My approach was comparing each answer choice:

A) (2^1/2)^6 = 2^3 = 8, whereas (6^1/6)^6 = 6, so A) is greater than E).

B) (3^1/3)^6 = 3^2 = 9, so B) is greater than A).

Now A) is equal to C) since (2^1/2)^4 = 4 and (4^1/4)^4 = 4, so C) is also smaller than B).

Finally, we compare A) to D) by raising to the power of 10: (2^1/2)^10 = 2^5 = 32 and (5^1/5)^10 = 25, so D) is smaller than A), and since A) is smaller than B), D) is also smaller than B).

We can conclude that B) is the biggest number.

Which of the following quantities is the largest?

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GMAT Problem Solving (PS) Questions

Which of the following quantities is the largest?

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