Is it possible to simplify equations like the following

[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?

 

[micromass]

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.

 

[chiro]

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)

 

[Mentallic]

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.

 

[epenguin]

You can factorise them, using the ordinary rules of numbers raised to powers, e.g. the first would be
12^x(5^x - 3^x) . 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.

 

[gokuls]

Merci beaucoup à tout! I suspected that it wouldn't be able to reducible, but I just wanted to make sure.

 

[HallsofIvy]

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?

 

[pierce15]

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.

 

[D H]

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.

 

[D H]

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 [strike]equations[/strike] 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 1^x=1:

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