Which of the following quantities is the largest?

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

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.

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!

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 \(3^{20} > 5^{12}\)

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

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