Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $6^{2n+2}$ ?

share|improve this question
    
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

3 Answers 3

up vote 0 down vote accepted

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 \ . $

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

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

share|improve this answer
    
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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.