Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
The underlying math facts that you need to know for this problem are the powers of 2, through 210. Know these powers, or be able to rederive them.
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256
29 = 512
210 = 1,024
We are asked what integer power of 2 gives us a cube root close to 50. In other words, 503 should be close to that power of 2.
In equation form, we have 2n ≈ 503. What is n?
One approach is to find 503, which equals (53)(103) = 125 × 1,000 or 125,000.
Look at our list of powers of 2 up to 210, and let’s match up to 125,000.
125 is approximately 27 (= 128), while 1,000 is approximately 210 (= 1,024).
So 125,000 is approximately 27 × 210, or 217. So n = 17. No other power of 2 is even close to 125,000. By the way, 217 equals 131,072, but the last thing you should do is calculate that number exactly.
You could also take the original equation and multiply in 23:
2n ≈ 503
23 × 2n ≈ 23 × 503
2n+3 ≈ 1003 = 1,000,000 = 1,0002.
Since 210 ≈ 1,000, we know that 1,0002 ≈ (210)2 = 220.
So n + 3 = 20, or n = 17.
The correct answer is B.
