The Quotient Rule of Exponents and Negative Exponents

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SWBAT simplify exponential expressions using the Quotient Rule and rewrite expressions with negative exponents.

Big Idea

To use quotients to show why any expression to the Zero Power is 1! and to clear up the misconception of dividing the exponents instead of subtracting them.

Warm Up

10 minutes

The goal of this lesson is for students to be able to simplify exponential expressions by:

  • expanding, and dividing factors
  • using the quotient property of exponents 
  • simplifying factors with negative exponents
  • rewrite expressions with negative exponents 

The Warm up takes about ten minutes for the students to complete and for me to use to introduce the lesson.  Students' prior knowledge from the previous lesson on expanding exponential expressions based on the meaning of an exponent can be used in these problems. After expanding, students make the decision to write the factors as one factor by adding the exponents, or simplifying any factors that divide to one. 

I want the students to build a strong foundation using the Shortcut Rule for Products, Powers, and Quotients. We will use this approach with simpler expressions before simplifying more complicated expressions in the next lesson using the Power Rule.  I review the Warm up in the video below.


Guided Notes

15 minutes

After I use the Warm Up to introduce the lesson, I have students complete geometric sequences in the Guided Notes.  Students then use the patterns to complete the tables. The students understand how to complete the patterns easily by deciding what factor to multiply by each output value.  After students have completed the tables, I want them to observe what is happening as the exponents decrease in value.   The tables make it easy for students to visualize that any number to the zero power is equal to one, and that negative exponents create a fraction.  

Students then are able to complete the rule that any factor to a zero power is equal to one.  As well as the rule that to rewrite a negative exponent as a positive exponent the location of the factor must be changed to a different location, which could be the numerator or the denominator.   

At the end of the Guided Notes, students compare the two methods of using The Quotient Rule Of Exponents , and dividing like factors to one.  In both methods students use the fraction bar to divide the numbers, or reduce the factors, and then simplify the variables by either subtracting the exponents, or expand to divide like factors. I ask students to provide reasoning of which method that they prefer because my intent is for each student to summarize one of the methods in their reasoning of why they prefer that method.

Independent Practice

15 minutes

The Guided Notes help provide students a good understanding of the Zero Rule of Exponents and the rules of working with negative exponents before working individually on the Independent Practice.  I time students about ten to fifteen minutes to complete the Independent Practice. Then, we review the work as a class.  During the review students self-grade their own work.  This immediate feedback helps to reinforce the objective of the lesson.   I plan to take questions, and model problems for students as needed.   

After students complete the Practice, I will allow them to start working on the Exit Slip as they complete the Independent Practice while other students are still working to complete the Independent Practice.


Exit Slip

10 minutes

In this Exit Slip my goal is to give students an opportunity to show reasoning (MP2) as tjeu simplify exponential expressions.  I will instruct students to write correct or incorrect on each of the four problems, but then to show their work and the reasoning of why the work is correct or incorrect. My students will hand in the Exit Slip when they complete it. I use this Exit Slip as a quick formative assessment to check for understanding of the Quotient Rule of Exponents and how to simplify expressions with negative exponents.

Exercise 1 on the Exit Slip shows a common mistake by students working with exponents which is to divide the exponents instead of subtracting the exponents of like bases.

Exercise 2 on the Exit Slip has the correct answer of 1, but I want students to show the work of why it is correct.  The student work should demonstrate that y to the fourth divided by y to the fourth is equivalent to y to the zero power which equals one.

Exercise 3 on the Exit Slip deals with the student understanding the reasoning behind why certain factors are in the numerator or denominator after simplifying.  This reasoning comes from a good foundation of numbers sense when working with whole numbers, integers, and rational expressions.  

Exercise 4 on the Exit Slip is similar to exercise three, the student should understand that the x is the only factor being taken to a power of negative one.  The three should remain in the numerator, and the x should be the only factor moved to the denominator.  Again understanding the meaning of the number based on its location in the numerator or denominator and which factors are being raised to a power.