1990 AMC 12 Problem 5

Attempt Problem 5 of the 1990 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1990 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Which of these numbers is largest?

\sqrt{\sqrt[3]{5\cdot6}}

\sqrt{6\sqrt[3]5}

\sqrt{5\sqrt[3]6}

\sqrt[3]{5\sqrt6}

\sqrt[3]{6\sqrt5}

Every choice is positive, so raise each one to the sixth power

Express each sixth power as

5^a6^b

Raising the five positive choices to the sixth power preserves their order and gives, respectively,

\begin{aligned} 30,\qquad &6^3\cdot5=1080,\\ 5^3\cdot6&=750,\\ 5^2\cdot6&=150,\qquad 6^2\cdot5=180. \end{aligned}

The largest is the second.

Thus the correct answer is B.

Problem 5 in other years