1970 AMC 12 Problem 13

Attempt Problem 13 of the 1970 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 1970 AMC 12 solutions, or check the answer key.

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13.

Given the binary operation

*

defined by

a*b=a^b

for all positive numbers

a

and

b.

Then for all positive

a,

b,

c,

n,

we have:

a*b=b*a

a*(b*c)=(a*b)*c

a*(b^n)=(a*n)*b

(a*b)^n=a*(bn)

None of these

Replace every

*

using the definition before comparing the two sides

For choice D, both sides simplify to a power with exponent

bn

For choice D,

\begin{aligned} (a*b)^n&=(a^b)^n=a^{bn},\\ a*(bn)&=a^{bn}. \end{aligned}

Thus that identity always holds. The other choices would assert false general identities such as

a^b=b^a

a^{b^c}=a^{bc}.

Therefore, the correct answer is D.

Problem 13 in other years