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 $2^{6x+3}*4^{3x+6}=2^{6x+3}*(2^2)^{3x+6}=2^{6x+3}*2^{6x+12}=2^{12x+15}$. We also know that $8^{4x+5}=(2^3)^{4x+5}=2^{12x+15}$. Thus there are infinite solutions to this equation since $2^{12x+15}$ is always equal to $2^{12x+15}$. So the answer is $\text{greater than}$ $3$ $(E)$.

See also

1966 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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