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

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

So generally, you're looking at expressions of the form [itex]a^x-b^x[/itex] 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

[tex](6(10))^x- (6(6))^x= 6^x10^x- 6^x6^x= 6^x(10^x- 6^x)[/tex]

[tex](6(5))^x- (5(5))^x= 5^x6^x- 5^x5^x= 5^x(6^x- 5^x)[/tex]

However, neither [itex]10^x- 6^x[/itex] nor [itex]6^x- 5^x[/itex] can be further simplified.

You mean "where the x is the power."

piercebeatz

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

The first one can. The gcd of 10 and 6 is 2.

D H

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:

[tex]
\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}
[/tex]