Can somebody explain how to solve this?

[muzzyfatts]

so if you cant see it clearly, it's : 2^4*3^7 = 6^m*3^k = 6^p*2^s. What's the method to solving this?

 

[HallsofIvy]

6= 2*3 so factor 2^4*3^7= 2^4*3^(4+ 3) as (2^4*3^4)*3^3= 6^4*3^3.

Since the exponent on the "3" is greater than the exponent on the "2", in order to write it as "6^p*2^s" we need a negative exponent: 2^4*3^7= 2^(7- 3)*3^7= (2^7*3^7)*2^(-3)= 6^7*2^(-3).

 

[Deleted member 4993]

View attachment 21863
so if you cant see it clearly, it's : 2^4*3^7 = 6^m*3^k = 6^p*2^s. What's the method to solving this?

Since there are four variables - m, k, p and s - what/which do you need to solve for?

 

[JayJay]

Can you assume m, k, I and s are all integers?

 

[Deleted member 4993]

View attachment 21863
so if you cant see it clearly, it's : 2^4*3^7 = 6^m*3^k = 6^p*2^s. What's the method to solving this?

Since 2 and 3 are relatively prime:

2⁴ * 3⁷ = 6^m * 3^k = 2^m * 3^k+m

from this you can derive two equations with two unknowns and solve for those.

Continue......

 

[muzzyfatts]

Can you assume m, k, I and s are all integers?

m,k,l and s are all to be given separate integer values, according to the task.

 

[Deleted member 4993]

Since 2 and 3 are relatively prime:

2⁴ * 3⁷ = 6^m * 3^k = 2^m * 3^k+m

from this you can derive two equations with two unknowns and solve for those.

Continue......

Did you follow this path?

 

[muzzyfatts]

Did you follow this path?

Not yet. I just came back and saw all these replies. I'm gonna sit down with the task later tonight or early tomorrow morning (CET). Will get back to you guys with my solution. Really appreciate you guys taking the time to help out. Obviously I'm no math wiz, so it's nice to have such helpful people contributing with their knowledge and help. Thanks to all of you.

 

[Steven G]

There is no need at all to assume or know that m, k, l and s are integers. Just work outthe problem and if they turn out to be integers then they are integers. Knowing ahead of time to look for integers will not change how to to this problem.

 

[Dr.Peterson]

There is no need at all to assume or know that m, k, l and s are integers. Just work out the problem and if they turn out to be integers then they are integers. Knowing ahead of time to look for integers will not change how to to this problem.

I'm not so sure of that. Did you try doing it that way?

 

[Steven G]

I'm not so sure of that. Did you try doing it that way?

Yes, I did solve the problem without assuming the variables are integers and I am positive that I got the correct answer. From Post #5 for example it is immediate that m = 4. What am I missing?

 

[JayJay]

Factorising into primes assumes that the exponents are integers. There are an infinite number of solutions if fractional exponents are allowed.

 

[jonah2.0]

Tequila induced speculation follows.
[MATH]2^{4}3^{7}=6^{m}3^{k}=6^{p}2^{s}[/MATH][MATH]2^{4}3^{7}=(2^{log_{2}6})^m3^{k}=(2^{log_{2}6})^p(3^{log_{3}2})^s[/MATH]

 

[Dr.Peterson]

Yes, I did solve the problem without assuming the variables are integers and I am positive that I got the correct answer. From Post #5 for example it is immediate that m = 4. What am I missing?

You're assuming integers without admitting it.

Since 2 and 3 are relatively prime:

2⁴ * 3⁷ = 6^m * 3^k = 2^m * 3^k+m

from this you can derive two equations with two unknowns and solve for those.

Continue......

Let m be any real number; then

2^m * 3^k+m = 2⁴ * 3⁷​

6^m * 3^k = 2⁴ * 3⁷​

3^k = 2⁴ * 3⁷ / 6^m​

k = log₃(2⁴ * 3⁷ / 6^m)​

 

[Steven G]

You're assuming integers without admitting it.

Let m be any real number; then

2^m * 3^k+m = 2⁴ * 3⁷​

6^m * 3^k = 2⁴ * 3⁷​

3^k = 2⁴ * 3⁷ / 6^m​

k = log₃(2⁴ * 3⁷ / 6^m)​

Very interesting! You can write for example 2^p with base 3.

 

[Dr.Peterson]

Very interesting! You can write for example 2^p with base 3.

Yup.

Of course, we know from post #6 that the variables do have to be integers, as one would expect in such a problem; but it's good to remind students (or ourselves) that that needs to have been stated. The original post was incomplete.