## Is it possible to simplify equations like the following

These are just some example equations:

60^x-36^x
or
30^x-25^x

where the x is raised to the power. How can (if possible) I simplify these equations?

Blog Entries: 8
Recognitions:
Gold Member
Staff Emeritus
 Quote by gokuls These are just some example equations: 60^x-36^x or 30^x-25^x where the x is raised to the power. How can (if possible) I simplify these equations?
I don't see any equations. An equation should have an "=" sign somewhere.
 If you treat it as an expression (and not an equation like micromass pointed out), you might want to consider that for x > 0, y > 0, (SQRT(x))^(2a) - (SQRT(y))^(2b) = (SQRT(x)^a + SQRT(y)^b)*(SQRT(x)^a - SQRT(y)^b)

Recognitions:
Homework Help

## Is it possible to simplify equations like the following

 Quote by gokuls These are just some example equations: 60^x-36^x or 30^x-25^x where the x is raised to the power. How can (if possible) I simplify these equations?
So generally, you're looking at expressions of the form $a^x-b^x$ for positive a and b.
Sadly, no. That is the simplest form you can have it in.
 Recognitions: Gold Member You can factorise them, using the ordinary rules of numbers raised to powers, e.g. the first would be 12x(5x - 3x) . Whether you call that a simplification and whether and when it is of any usefulness are other questions, but it shouldn't be a difficulty to see.
 Merci beaucoup à tout! I suspected that it wouldn't be able to reducible, but I just wanted to make sure.

Recognitions:
Gold Member
Staff Emeritus
 Quote by gokuls These are just some example equations: 60^x-36^x
$$(6(10))^x- (6(6))^x= 6^x10^x- 6^x6^x= 6^x(10^x- 6^x)$$

 or 30^x-25^x
$$(6(5))^x- (5(5))^x= 5^x6^x- 5^x5^x= 5^x(6^x- 5^x)$$

However, neither $10^x- 6^x$ nor $6^x- 5^x$ can be further simplified.

 where the x is raised to the power.
You mean "where the x is the power."

 How can (if possible) I simplify these equations?

 Quote by gokuls These are just some example equations: 60^x-36^x or 30^x-25^x where the x is raised to the power. How can (if possible) I simplify these equations?
If you wanted to solve an equation in this form (e.g. set it equal to something like a constant) you could solve it with the Lambert W function.

Mentor
 Quote by HallsofIvy However, neither $10^x- 6^x$ nor $6^x- 5^x$ can be further simplified.
The first one can. The gcd of 10 and 6 is 2.

Mentor
 Quote by gokuls These are just some example equations: 60^x-36^x or 30^x-25^x where the x is raised to the power. How can (if possible) I simplify these equations expressions?
Previous posts have simplified these by factoring out the greatest common denominators of 60, 36 and of 30, 25.

Another way is to take advantage of the fact that 1x=1:

\begin{aligned} 60^x-36^x =& 36^x \left( (60/36)^x - (36/36)^x \right) = 36^x \left( (5/3)^x - 1\right) \\ 30^x-25^x =& 25^x \left( (30/25)^x - (25/25)^x \right) = 25^x \left( (6/5)^x - 1\right) \end{aligned}