Multiply and Divide Rational Exponents

Definition: When multiplying and dividing rational exponents, the following rules can be used (where m and n are rational numbers):

png.latex-%5Cinline %5Cdpi%7B150%7D %5Cbg_white %5C%5Cx%5Em%5Ccdot x%5En %3D x%5E%7Bm+n%7D%5C%5C%5C%5C%28x%5Em%29%5En %3D x%5E%7Bmn%7D%5C%5C%5C%5C%28xy%29%5Em %3D x%5Emy%5Em%5C%5C%5C%5Cx%5E%7B-m%7D %3D%5Cfrac%7B1%7D%7Bx%5Em%7D%5C%5C%5C%5C%5Cfrac

Example: Simplify

png.latex-%5Cinline %5Cdpi%7B150%7D %5Cbg_white %5Cfrac%7B125%5E%7B%5Cfrac%7B2%7D%7B9%7D%7D%5Ccdot 125%5E%5Cfrac%7B1%7D%7B9%7D%7D%7B5%5E%5Cfrac%7B1%7D%7B4%7D%7D.png

Solution: 

Step 1: Simply the numerator . As these terms have common bases, we can add their exponents and rewrite this as:

png.latex-%5Cinline %5Cdpi%7B150%7D %5Cbg_white %5C%21 %5C%21 %5C%21 %5Cfrac%7B125%5E%7B%5Cfrac%7B2%7D%7B9%7D%7D%5Ccdot 125%5E%5Cfrac%7B1%7D%7B9%7D%7D%7B5%5E%5Cfrac%7B1%7D%7B4%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B125%5E%7B%28%5Cfrac%7B2%7D%7B9%7D+%5Cfr.png

Step 2: Simplify the numerator further and divide

png.latex-%5Cinline %5Cdpi%7B150%7D %5Cbg_white %5C%5C%5C%5C%5Cfrac%7B5%7D%7B5%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%7D%3D%5Cfrac%7B5%5E1%7D%7B5%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%7D%3D5%5E1 - 5%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D5%5E%7B%281-%5Cfrac%7B1%7D%7B4%7D%29.png

 

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