# Is it possible to simplify equations like the following

1. Dec 26, 2012

### 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?

2. Dec 26, 2012

### micromass

Staff Emeritus
I don't see any equations. An equation should have an "=" sign somewhere.

3. Dec 27, 2012

### 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)

4. Dec 27, 2012

### Mentallic

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.

5. Dec 27, 2012

### epenguin

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.

6. Dec 27, 2012

### gokuls

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

7. Dec 28, 2012

### HallsofIvy

Staff Emeritus
$$(6(10))^x- (6(6))^x= 6^x10^x- 6^x6^x= 6^x(10^x- 6^x)$$

$$(6(5))^x- (5(5))^x= 5^x6^x- 5^x5^x= 5^x(6^x- 5^x)$$

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

You mean "where the x is the power."

8. Jan 3, 2013

### pierce15

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.

9. Jan 3, 2013

### D H

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

10. Jan 3, 2013

### D H

Staff Emeritus
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}