SOLUTION: 216^-2k -------- = 36 Please show steps and solution. 36^-2k-3

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Question 1135939: 216^-2k
-------- = 36 Please show steps and solution.
36^-2k-3
Found 4 solutions by Alan3354, Theo, MathTherapy, greenestamps:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
What is the exponent in the DEN?
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
well, .....

i got k = 2 and it appears to be correct.

here's my worksheet.

$$$
$$$

in step 1 i copied the problem.

in step 2 i multiplied both sides of the equation by 36^(-2k-3)

in step 3 i converted 36^(-2k-3) to 36^(-2k) / 36^3 because they're equivalent.

in step 4 i divided 36 in the numerator into 36^3 in the denominator to get 36^2 in the denominator.

in step 5 i replaced (6 * 36)^(-2k) with 6^(-2k) * 36^(-2k) because they're equivalent.

in step 6 i divided both sides of the equation by 36^(-2k) which then removed 36^(-2k) from the equation because 36^(-2k) / 36^(-2k) = 1.

in step 7 i made 6^(-2k) equal to 1 / 6^(2k) because they're equivalent.

in step 8 i cross multiplied.

in step 9 i took the log of both sides of the equation to get log(36^2)( = log(6^(2k)) which then became 2 * log(36) = 2k * log(6) because log(x^a) = a*log(x).

in step 10 i divided both sides of the equation by log(6).

in step 11 i divided both sides of the equation by 2.

in step 12 i solved for k.

any questions how or why i got what i got, send email to dtheophilis@gmail.com.

in donfirmed by replacing k with 2 in the original equation and evaluated it to get 36 = 36.


Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
216^-2k
-------- = 36 Please show steps and solution.
36^-2k-3
If , then:





 ----- Applying 

- 6k - (- 4k - 6) = 2 ------- Bases are equal and so are the exponents

- 6k + 4k + 6 = 2

- 2k = 2 - 6

- 2k = - 4



You DON'T NEED LOGS or any complex calculations....just the laws of exponents! 

Using logs is just a waste of time and effort, unless you're requested to use logs, of course. 

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!
 





There are of course many paths to solving this equation....


With the negative exponents in both numerator and denominator, I would first switch them to make the exponents positive:





Then, with bases 36=6^2 and 216=6^3 as the bases for the exponents, change every term in the equation to have base 6:











The bases are the same, so the exponents are equal:











 

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