Question 22390
<pre>Show each step in simplifying the following rational expressions. 
<pre>Also explain what a simplified rational expression looks like 
(how you know it is simplified). 
-15x^3·y^3/-20x·y^4 
<b><font size = 3>
-15x<sup>3</sup>y<sup>3</sup>
———————
 20xy<sup>4</sup>

Since 5 divides evenly into both -15 and 20, we divide each of these
coefficients by 5:

 <sub>-3</sub>
<s>-15</s>x<sup>3</sup>y<sup>3</sup>
———————
 <s>20</s>xy<sup>4</sup>
 <sup> 4</sup> 

-3x<sup>3</sup>y<sup>3</sup>
———————
 4xy<sup>4</sup>

Give the x in the bottom the exponent of 1

-3x<sup>3</sup>y<sup>3</sup>
———————
 4x<sup>1</sup>y<sup>4</sup>

Now the rule is:

When dividing exponentials with like bases, subtract the exponents
(larger minus smaller) and put the result in place of the exponential
which had the larger exponent, and eliminate the exponential with the 
smaller exponent.

Using the above rule:
x<sup>3</sup> and x<sup>1</sup> have like bases so we subtract exponents, larger minus smaller,
3 - 1, getting 2, and place x<sup>2</sup> in place of x<sup>3</sup>, then eliminate the x<sup>1</sup> 

-3x<sup>2</sup>y<sup>3</sup>
———————
  4y<sup>4</sup>

Using the above rule again:

y<sup>3</sup> and y<sup>4</sup> have like bases so we subtract exponents, larger minus smaller,
4 - 3, getting 1, and place y<sup>1</sup> in place of y<sup>4</sup>, then eliminate the y<sup>3</sup>:

 -3x<sup>2</sup>
———————
  4y<sup>1</sup>
 
Now we eliminate the 1 exponent of y<sup>1</sup>

 -3x<sup>2</sup>
———————
  4y

If we like we can also move the negative sign from the numerator out
in front of the whole fraction.

   3x<sup>2</sup>
— ——————
   4y

We know this is simplified because there are no common factors of the
numerator and denominator (other than 1), and also because there are no
exponentials with like bases.

Edwin
AnlytcPhil@aol.com</pre>