Is it possible to simplify equations like the following

  • Thread starter gokuls
  • Start date
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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?
 
21,993
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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

Science Advisor
4,783
127
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

Homework Helper
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94
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 [itex]a^x-b^x[/itex] for positive a and b.
Sadly, no. That is the simplest form you can have it in.
 

epenguin

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

HallsofIvy

Science Advisor
Homework Helper
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These are just some example equations:

60^x-36^x
[tex](6(10))^x- (6(6))^x= 6^x10^x- 6^x6^x= 6^x(10^x- 6^x)[/tex]

or
30^x-25^x
[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.

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

How can (if possible) I simplify these equations?
 
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

Staff Emeritus
Science Advisor
Insights Author
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680
However, neither [itex]10^x- 6^x[/itex] nor [itex]6^x- 5^x[/itex] can be further simplified.
The first one can. The gcd of 10 and 6 is 2.
 

D H

Staff Emeritus
Science Advisor
Insights Author
15,329
680
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 1x=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]
 

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