1966 AMC 12 Problem 12

Attempt Problem 12 of the 1966 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 1966 AMC 12 solutions, or check the answer key.

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12.

The number of real values of

x

that satisfy the equation

(2^{6x+3})(4^{3x+6})=8^{4x+5}

is:

0

1

2

3

greater than

3

Rewrite every term with base

2

Compare the total exponent on each side before trying to solve for

x

In base

2,

the left exponent is

\begin{aligned} &(6x+3)+2(3x+6)\\ &\qquad=12x+15, \end{aligned}

and the right exponent is

3(4x+5)=12x+15.

The equation is an identity for every real

x,

so it has more than three real solutions.

Therefore, the correct answer is E.

Problem 12 in other years