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 
Resources 

RED BOOK Reference:
Pages: 417 ─ 421
Practice on page 421,
#5, 7, 9, 11, 1521 odds,
and 2935 odds. 
BLACK BOOK Reference:
Pages: 440 ─ 445
Practice on page 443,
#1323 odds. 
Lesson Vocabulary   Exponents  A 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). 

Strategies and Content Practice 
Begin (prepare) 
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 8^{3} is smaller than 8^{7} so the 8^{7} stays in the denominator and the 8^{3} 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 8^{4}. 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 Q^{6} is bigger than Q^{2} so the Q^{6} stays in the numerator and the Q^{2} goes away. The Q^{2} is replaced with a "1". The new exponent value is "Big exponent ─ Small exponent" or (6 ─ 2) = 4. So the new numerator is Q^{4}. The same is true with the fraction F^{3} over F^{7}. The F^{7} is a bigger value than F^{3}. So when we simplify, the F in the denominator stays and the F^{3} 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 F^{4}.
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 standalone 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 B^{8} is bigger than B^{3}, the B in the numerator stays and the B in the denominator goes away. The B^{3 }does 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 B^{5}.

Assess (formative) 
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. 
