<PRE><B>Question 62417</B>
&lt;pre&gt;Hi,
Can you please show me the steps needed to solve this?

((3^-2)^a).((3^b)^(-2a))
----------------------
((3^-2)^b).((9^-b)^(3a))

three to the negative two to the a times three to the
b to the negative two a divided by
three to the negative two to the b times nine to the 
negative b to the three a

Thanks in advance!!

Ray
&lt;font size = 5&gt;&lt;b&gt;
 (3&lt;sup&gt;-2&lt;/sup&gt;)&lt;sup&gt;a&lt;/sup&gt;(3&lt;sup&gt;b&lt;/sup&gt;)&lt;sup&gt;-2a&lt;/sup&gt;
---------------
 (3&lt;sup&gt;-2&lt;/sup&gt;)&lt;sup&gt;b&lt;/sup&gt;(9&lt;sup&gt;-b&lt;/sup&gt;)&lt;sup&gt;3a&lt;/sup&gt;

First multiply the outer exponents by the inner 
exponents to remove the parentheses:

 3&lt;sup&gt;-2a&lt;/sup&gt;3&lt;sup&gt;-2ab&lt;/sup&gt;
-----------
 3&lt;sup&gt;-2b&lt;/sup&gt;9&lt;sup&gt;-3ab&lt;/sup&gt;

Rewrite the 9 as (3&lt;sup&gt;2&lt;/sup&gt;)

   3&lt;sup&gt;-2a&lt;/sup&gt;3&lt;sup&gt;-2ab&lt;/sup&gt;
--------------
 3&lt;sup&gt;-2b&lt;/sup&gt;(3&lt;sup&gt;2&lt;/sup&gt;)&lt;sup&gt;-3ab&lt;/sup&gt;

Remove the parentheses by multiplying the outer 
exponent -3ab by the inner exponent 2, getting
-6ab:

 3&lt;sup&gt;-2a&lt;/sup&gt;3&lt;sup&gt;-2ab&lt;/sup&gt;
-----------
 3&lt;sup&gt;-2b&lt;/sup&gt;3&lt;sup&gt;-6ab&lt;/sup&gt;

Get rid of each negative exponent by using the 
rules:

1. To get rid of a negative exponent in the 
   numerator, move the base and exponent from 
   the numerator to the denominator and
   change the sign of the exponent to positive.

2. To get rid of a negative exponent in the 
   denominator, move the base and exponent from 
   the denominator to the numerator and
   change the sign of the exponent to positive.

So move the 3&lt;sup&gt;-2a&lt;/sup&gt; from the top to the bottom as 3&lt;sup&gt;2a&lt;/sup&gt;,
move the 3&lt;sup&gt;-2ab&lt;/sup&gt; from the top to the bottom as 3&lt;sup&gt;2ab&lt;/sup&gt;,
move the 3&lt;sup&gt;-2b&lt;/sup&gt; from the bottom to the top as 3&lt;sup&gt;2b&lt;/sup&gt;, and
move the 3&lt;sup&gt;-6ab&lt;/sup&gt; from the bottom to the top as 3&lt;sup&gt;6ab&lt;/sup&gt;


 3&lt;sup&gt;2b&lt;/sup&gt;3&lt;sup&gt;6ab&lt;/sup&gt;
---------
 3&lt;sup&gt;2a&lt;/sup&gt;3&lt;sup&gt;2ab&lt;/sup&gt;

Now on the top we add the exponents of 3, which 
are 2b and 6ab, getting 2b+6ab 

On the bottom we also add the exponents of 3, 
which are 2a and 2ab, getting 2a+2ab

 3&lt;sup&gt;2b+6ab&lt;/sup&gt;
--------- 
 3&lt;sup&gt;2a+2ab&lt;/sup&gt;

Finally we subtract exponents (2b+6ab)-(2a+2ab)

3&lt;sup&gt;(2a+6ab)-(2a+2ab)&lt;/sup&gt;

3&lt;sup&gt;2b+6ab-2a-2ab&lt;/sup&gt;

3&lt;sup&gt;2b-2a+4ab&lt;/sup&gt;

or if you like, factor out
2 in the exponent:

3&lt;sup&gt;2(b-a+2ab)&lt;/sup&gt;

Edwin&lt;/pre&gt;



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