# Problem of the Day

If , what is the value of ?

A. 2

B. 3

C. 6

D. 12

E. 18

### Solution

[latexpage]

The key to this problem is to recognize that $0.375=\frac{3}{8}$. Then you can proceed in the following manner:

$0.375^3=2^a3^b6^c$

$\displaystyle (\frac{3}{8})^3=2^a3^b6^c$

$\displaystyle \frac{3^3}{8^3}=2^a3^b6^c$

$3^38^{-3}=2^a3^b6^c$

$3^3(2^3)^{-3}=2^a3^b(2\cdot 3)^c$

$3^32^{-9}=2^a3^b2^c3^c$

$2^{-9}3^3=2^{a+c}3^{b+c}$

$a+c=-9$ and $b+c=3$

$(b+c)-(a+c)=3- -9$

$b-a=12$

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