# Help with finding value of exponents?

Find the value of n:

$54=2^n*3^{n+2}$

I don't know where to start. can we times 2 and 3 and get $6^{2n+2}$ ?

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You can test your last question. Try it with $n=1$ and see if you get the same thing before and after. –  Eric Towers Mar 9 at 3:12
$54 = 2\times 27$ and guess. –  John Mar 9 at 3:13

Remember that $\ a^{n+2} \ = \ a^n \ \cdot \ a^2 \ .$ You can write the right-hand side of your equation as $\ 2^n \ \cdot \ 3^n \ \cdot \ 3^2 \ .$ And you can say that $\ 2^n \ \cdot \ 3^n \ = \ (2 \cdot 3)^n \ = \ 6^n \ .$

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Right, I forgot that $a^{n+2} = a^n ⋅ a^2$ Thanks! –  Helena Mar 9 at 3:57

$$54=2^n*3^{n+2}=2^n*3^n*3^2=9(2^n*3^n)$$ $$\implies 6=2^n*3^n=(2*3)^n=6^n$$ $$\implies n=log_6(6)= 1$$

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Oops, thanks for the edit, John! –  WChang Mar 9 at 3:57

You know because the number (54) is so low that the value of n will also be a relatively low number. You also know that 54 is 27 * 2, and 27 is 3^3.

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