1985 AMC 12 Problem 15

Attempt Problem 15 of the 1985 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 1985 AMC 12 solutions, or check the answer key.

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

15.

a

and

b

are positive numbers such that

a^b=b^a

and

b=9a,

then the value of

a

9

\frac19

\sqrt[9]{9}

\sqrt[3]{9}

\sqrt[4]{3}

Substitute

b=9a

into

a^b=b^a

Because

a\gt0,

take the

a

-th root and simplify

Substitution gives

a^{9a}=(9a)^a.

Taking the positive

a

-th root yields

a^9=9a,

a^8=9.

Therefore

a=9^{\frac{1}{8}}=3^{\frac{1}{4}}=\sqrt[4]{3}.

Thus the correct answer is E.

Problem 15 in other years