M01-08

Official Solution:

Which of the following expressions has the greatest value?

A. \(1^{999}\)
B. \(2^{300}\)
C. \(3^{200}\)
D. \(4^{100}\)
E. \(16^{50}\)


To compare the given expressions effectively, consider a common base or power. Observe that all expressions can be expressed as some power of 100:

A. \(1^{999}\). Taking 1 to any power is still 1. If the base were a different number, then we could approximate the answer by using a power of 1,000.

B. \(2^{300} = 2^{3*100}=(2^3)^{100} = 8^{100}\).

C. \(3^{200} =3^{2*100} = (3^2)^{100} = 9^{100}\).

D. \(4^{100} = 4^{100}\).

E. \(16^{50} = (4^2)^{50} = 4^{100}\).

Therefore, option C, \(3^{200}\), has the greatest value.


Answer: C