Dividing (Simplifying) Exponents
 GOAL: I can divide or simplify fractions that contain numbers or variables that have exponents

 Learning Standards  Common Core Standard: 7.EE.1


RED BOOK Reference:

Pages: 417 ─ 421

Practice on page 421,

#5, 7, 9, 11, 15-21 odds,
and 29-35 odds.
BLACK BOOK Reference:
Pages: 440 ─ 445

Practice on page 443,
#13-23 odds.

 Practices (with answers)  

 Lesson Vocabulary 
  ExponentsA number written above and to the right of another number, variable, or mathematical expression that shows how many times the number, variable, or expression is multiplied by itself.
 Monomials Monomials are the small units or pieces of algebra expressions.  They can be as small as a single number(  2, ─7), or a variable (x, y, a), or a number and variable combination (3d, ─8g, 4x²).

 Video Tutorials  
TopicDescription Time (min: sec)
Exponent Properties 1 Khan Academy2:36
Exponent Properties 2   Khan Academy5:12 
Exponent Properties 3   Khan Academy2:35
Exponent Properties 4   Khan Academy3:07

Strategies and Content Practice 
Numbers and variables that have exponents can be operated on by the usual math operations like addition, subtraction, multiplication and division.  This lesson covers dividing exponents.
Engage As defined above, monomials are small expressions of an algebra problem.
These monomials may contain exponents like:

When we divide two or more numbers with exponents, we start by identifying the base number that has the larger exponent.  The base number (or variable as we'll see later) says in its position (numerator or denominator) and the other one goes away.  Then the new exponent value of the remaining base number is calculated by SUBTRACTING the smaller exponent from the bigger exponent.  For example:

Above, the 83 is smaller than 87 so the 87 stays in the denominator and the 83 goes away.  In this case it is replaced with a "1" because we cannot leave the numerator or the denominator of a fraction blank.  The new exponent value is "Big exponent ─ Small exponent" or (7 ─  3) = 4.  So the new denominator is 84.
The same is true with the fraction (─6)5 over (─6)2.  The (─6)5 is a bigger value than (─6)2.  So when we simplify, the (─6) in the numerator stays and the (─6)2 goes away and is replaced by "1".  The new exponent value for the (─6) in the numerator is "Big exponent ─ Small exponent" or (5 ─ 2) = 3.  So the new numerator is (─6)3.    

When we divide two or more variables with exponents, the same rules apply as above.  For example:

Above, the Q6 is bigger than Q2 so the Q6 stays in the numerator and the Q2 goes away.  The Q2 is replaced with a "1".  The new exponent value is "Big exponent ─ Small exponent" or (6 ─  2) = 4.  So the new numerator is Q4.
The same is true with the fraction F3 over F7.  The F7 is a bigger value than F3.  So when we simplify, the F in the denominator stays and the F3 goes away and is replaced by "1".  The new exponent value for the F in the denominator is "Big exponent ─ Small exponent" or (7 ─ 3) = 4.  So the new denominator is F4.

When numbers are with variables, we call them coefficients.
Monomials with coefficients and variables can be divided (simplified).
To do this, we first simplify the coefficients (if possible) as if they were stand-alone fractions.
Then we simplify the variables is the way discussed above.
For example:

First we simplify the coefficients 10/6.  Both divide by 2 so they simplify to 5/3.

Then we simplify the B variables.  
Since the B8 is bigger than B3, the B in the numerator stays and the B in the denominator goes away.  The Bdoes not need to be replaced with a "1" because there is already a "3" in the denominator from the simplified coefficients.  The new exponent value is "Big exponent ─ Small exponent" or (8 ─  3) = 5.  So the new variable in the numerator is B5.

After example problems and videos, try to solve the Practice Worksheets above.  The answers to the practices are on the last page so you can check how well you are doing.

Please see me for more help if you are having difficulty with the practices.