# Quotient of Powers

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## Objective

SWBAT develop and apply the Quotients of Powers Property

#### Big Idea

Students explore, then apply the Quotient of Powers property.

## Launch

15 minutes

At the closure of the previous lesson on the Power of a Power Property, students were asked to write 2 key ideas they learned in the lesson. Call on a couple of students to share what they wrote. I would expect students to mention things relative to the use of the repeated multiplication model for powering, using the product of powers to demonstrate power of a power, or being able to simplify expressions involving both power of a power and multiplication. After a few students share what they wrote, project the following two tasks on the board for each individual student to answer in their notebooks: 2 Step Launch.docx

Motivate students to use a calculator to complete the tasks quicker. This will also free up time for them to focus on the pattern and the meaning of Question 1.

Students should list the following values for step 1:

3, 9, 27, 81, 243, 729, and 2187

Students may be surprised to see that the quotient when dividing the two chosen numbers will always be on the list. Make sure students divide a larger by a smaller number to avoid decimals since they have not had negative exponents.

Then, allow students to show and discuss their work quietly with a neighboring student.  Some students may well have chosen different values to divide. Make certain that the students answer Step 2 and have a couple of students share their explanation aloud.

The explanation we want is something along the lines of "dividing the powers yields a quotient that is also a result of repeatedly using 3 as a factor in the numerator (dividend) and in the denominator (divisor). It is always a good idea to encourage students to use correct terminology. So, enforce precise language as each student gives his/her explanation.

## New Info

15 minutes

In this section of the lesson, project NEWINFO QUOTIENT OF POWERS on the whiteboard.

I recommend asking students to try these independently, allowing no more than 10 minutes to answer the 3 problems in part I, and the question in part II.

In part I, make certain that students are both expanding the powers and simplifying to obtain a power with a positive exponent.  Some students may not see that when cancelling the x’s the answer is 1. Make them write in the ones as place holders. See student Quotient of powers.jpg for an example what we want to see students do. Have 3 students come to the board and show their work.

Part II asks students to compare the exponents of the quotient with those of the dividend and divisor powers. Ask a student to write a general rule from this pattern, using variables. Have a student share the rule they came up with and write this rule on the board. State then, that this is the Quotient of Powers Property of exponents.

In part III, students should see that the answer is 1. They can either expand the powers and show that all terms cancel, or recognize that a non-zero number divided by itself is 1.     Make sure they see this both ways.

Show students that applying the Quotient of Powers Property here shows that n0 = 1

Ask students to show this with a different example and then state that this is the Zero Exponent Property.

## Application

20 minutes

Hand each student a copy of the APPLY QUOTIENT PROPERTY HANDOUT. There are 5 questions where students must apply the Quotient of Powers property. Students can work independently or in pairs.

For question 1, students must use scientific notation. A quick refresher on this may help if time permits. In questions 2 and 3, students may subtract the coefficients instead of dividing. This is serious and these students may need to expand the powers, simplify, and analyze what happens with the exponents.

Question 5 is a reminder that the identical bases are needed to apply the property. Guide students to see that either the numerator or denominator can be changed to make the bases the same.

## Closure

5 minutes

To end the lesson, write the following expression on the board: 2a2b

Ask students to write an algebraic fraction that can be simplified using the Quotient of Powers Property to yield this expression.

Call on students and write a few of the answers on the board.

## EXTENSION

A good extension problem could be to simplify a fraction which includes having to use the 3 properties covered in this unit so far, and include negative exponents in the denominator. Although the Negative Exponent Property has not been given, they could be given a fraction which will simplify into positive exponents.

Click EXTENSION EXAMPLE for a sample problem.