Simplifying Expressions Using Exponent Laws.

Date: 02/08/98 at 13:32:54
From: Luke Bartley
Subject: Simplifying expressions using exponent laws

To start off, I am having much difficulty with these types of 
questions. One example is: (Simplify) 

5 to the exponent -4/3 divided by 5 to the exponent 2/3

The answer that was given to me was 1/25.
I was 2 steps into the question when I started having trouble. How do 
I solve this question?

Date: 02/25/98 at 09:47:28
From: Doctor Sonya
Subject: Re: Simplifying expressions using exponent laws

Hi Luke,

Simplifying exponential expressions like the one above is hard, but 
it gets easier with practice, I promise.  

I'll help you work through your example in two different ways.  
You wrote, "5 to the exponent -4/3 divided by 5 to the exponent 2/3."  
I'm going to write it as:

    5^(-4/3) / 5^(2/3)   (5^n means "5 to the nth power")

The first thing to take care of are the negative exponents.  You 
probably already know the general rule:

    a^(-x) = 1/a^x.

For example, 

    4^(-2) = 1/4^2 = 1/16
    3^(-3) = 1/3^3 = 1/127

Therefore you can rewrite your problem as:
    1/5^(4/3)
    ---------
     5^(2/3) 

You can simplify this compound fraction to:

          1
  -----------------
  5^(4/3) * 5^(2/3)

The * means multiply.  Make sure you are clear on how I did this step.

Now you can multiply the two terms in the denominator together. Since 
they are both "5 to the something", all you have to do is add the 
exponents.  When you do this, you'll get:

          1     
     -----------
     5^(4/3+2/3)

When I add the exponents together, I get:
          1
         ---
         5^2

There is another rule of exponents that can also help solve this 
problem.  Remember that:

    x^a/x^b = x^(a-b)

What this says is that when you divide, you subtract the exponents.  
It's the opposite of multiplication, where you add the exponents.  In 
our example, we had:

    5^(-4/3) / 5^(2/3) 
   
So if I follow the rule above, I'll get that it is equal to:

    5^(-4/3 - 2/3)

Simplifying this expression gives me:

    5^(-6/3) = 5^(-2)

In general, when you do these types of problems, keep the rules for 
adding, subtracting, multiplying, and dividing in mind (or next to you 
on a piece of paper) and see if you can apply any of them.  They will 
almost always allow you to simplify.

-Doctors Sonya and Fox,  The Math Forum
Check out our web site http://mathforum.org/dr.math/