The cube root of what integer power of 2 is closest to 50?

MANHATTAN GMAT OFFICIAL SOLUTION:

The underlying math facts that you need to know for this problem are the powers of 2, through 2^10. Know these powers, or be able to rederive them quickly.

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1,024

We are asked what integer power of 2 gives us a cube root close to 50. In other words, 50^3 should be close to that power of 2.

In equation form, we have 2^n ≈ 50^3. What is n?

One approach is to find 50^3, which equals (5^3 )(10^3 ) = 125 × 1,000 or 125,000.

Look at our list of powers of 2 up to 2^10, and let’s match up to 125,000.

125 is approximately 2^7 (= 128), while 1,000 is approximately 2^10 (= 1,024).

So 125,000 is approximately 2^7 × 2^10, or 2^17. So n = 17. No other power of 2 is even close to 125,000. By the way, 2^17 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 2^3:

2^n ≈ 50^3
2^3 × 2^n ≈ 2^3 × 50^3
2^(n+3) ≈ 100^3 = 1,000,000 = 1,000^2.

Since 2^10 ≈ 1,000, we know that 1,000^2 ≈ (2^10)^2 = 2^20.

So n + 3 = 20, or n = 17.

The correct answer is B.