1983 AHSME Problems/Problem 25

Problem 25

If $60^a=3$ and $60^b=5$ , then $12^{(1-a-b)/\left(2\left(1-b\right)\right)}$ is

$\textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 2\sqrt{3}\qquad$

Solution

We have that $12=\frac{60}{5}$ . We can substitute our value for 5, to get $12=\frac{60}{60^b}=60\cdot 60^{-b}=60^{1-b}.$ Therefore we have $12^{[(1-a-b)/2(1-b)]}=60^{[(1-b)(1-a-b)/2(1-b)]}=60^{(1-a-b)/2}.$ Since $4=\frac{60}{3\cdot 5}$ , we have $4=\frac{60}{60^a 60^b}=60^{1-a-b}.$ Therefore, we have $60^{(1-a-b)/2}=4^{1/2}=\fbox{\textbf{(B)} 2}$

1983 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Problem 26
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1983 AHSME Problems/Problem 25

Problem 25

Solution

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