MathDB
Binary Operations

Source: 1970 AHSME Problem 13

March 24, 2014
AMC

Problem Statement

Given the binary operation ∗\ast defined by a∗b=aba\ast b=a^b for all positive numbers aa and bb. The for all positive a,b,c,n,a,b,c,n, we have
<spanclass=′latex−bold′>(A)</span>a∗b=b∗a<spanclass=′latex−bold′>(B)</span>a∗(b∗c)=(a∗b)∗c<span class='latex-bold'>(A) </span>a\ast b=b\ast a\qquad<span class='latex-bold'>(B) </span>a\ast (b\ast c)=(a\ast b)\ast c\qquad
<spanclass=′latex−bold′>(C)</span>(a∗bn)=(a∗n)∗b<spanclass=′latex−bold′>(D)</span>(a∗b)n=a∗(bn)<spanclass=′latex−bold′>(E)</span>None of these<span class='latex-bold'>(C) </span>(a\ast b^n)=(a\ast n)\ast b\qquad<span class='latex-bold'>(D) </span>(a\ast b)^n=a\ast (bn)\qquad <span class='latex-bold'>(E) </span>\text{None of these}
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