1966 AHSME Problems/Problem 12

Problem

The number of real values of $x$ that satisfy the equation $(2^{6x+3})(4^{3x+6})=8^{4x+5}$ is:

$\text{(A) zero} \qquad \text{(B) one} \qquad \text{(C) two} \qquad \text{(D) three} \qquad \text{(E) greater than 3}$

Solution

We know that $2^{6x+3}\cdot4^{3x+6}=2^{6x+3}\cdot(2^2)^{3x+6}=2^{6x+3}\cdot2^{6x+12}=2^{12x+15}$ . We also know that $8^{4x+5}=(2^3)^{4x+5}=2^{12x+15}$ . There are infinite solutions to the equation $2^{12x+15}=2^{12x+15}$ , so the answer is $\boxed{\text{(E) greater than 3}}$ .

1966 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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1966 AHSME Problems/Problem 12

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