In this lesson, students will learn how to combine exponents when dividing or multiplying powers with the same base. Students are going to:
- discover how to combine exponents while multiplying powers with the same base.
- apply the rule for combining exponents while multiplying powers with the same base.
- discover how to combine exponents while dividing powers with the same base.
- apply the rule for combining exponents while dividing powers with the same base.
- How can mathematics help to quantify, compare, depict, and model numbers?
- Exponent: A numeral that tells how many times a number or variable is used as a factor. For example, in \(2^7\), 2 is the base and 7 is the exponent; this means 2 is multiplied by itself 7 times.
- Multiplying Powers worksheet (M-8-4-1_Multiplying Powers and KEY) for each student.
- Dividing Powers worksheet (M-8-4-1_Dividing Powers and KEY) for each student.
- Powers in Expressions packet (14 sheets) (M-8-4-1_Powers in Expressions) one copy or additional copies as needed.
- Students' completion of the Multiplying Powers worksheet may be used to evaluate them.
Based on how well they complete the Dividing Powers worksheet, teachers may assess the understanding of their students.
- The degree of student mastery will be ascertained through student performance and instructor observation during Activity 3 utilizing Powers in Expressions (M-8-4-1_Powers in Expressions).
Formative Assessment, Modeling, Scaffolding, Active Engagement
W: Students will be able to combine and simplify powers that involve either division or multiplication with the same base. Students will use a variety of sample problems to master these skills.
H: Students will become engrossed in the material by starting the course with basic questions (evaluating \(3^2\) and \(3^5\)). They will be able to conjecture and learn the method for simplifying equations when multiplying powers with the same base.
E: Following teacher-guided teaching, students will look at a number of examples and investigate simplifying them on their own using the Multiplying Powers and Dividing Powers worksheets. Students will delve deeper into important concepts in the class-based Task 3.
R: As they go over accurate and inaccurate assumptions and conclusions on the practice worksheets, students will polish and improve their comprehension of multiplying and dividing powers with like bases. They will also simplify outcomes. Lesson concepts will be reviewed through practice and evaluation of outcomes.
E: How well students complete the practice worksheet issues could be taken into account when evaluating them. Make necessary corrections and reteaches to guarantee that students comprehend the steps involved in multiplying and dividing powers using like bases. Comparing responses from classmates on Activity 3 will also assist students in assessing their own development.
T: Adapt the lesson to the student's requirements by using the Extension section. Throughout the year, there are options for reviewing course concepts under the Routine area. Activities for small groups are provided for students who can benefit from more practice. Students who are up to the challenge of surpassing the standards required are suited for the Expansion segment.
O: Students are allowed to practice both of the two distinct concepts—multiplying and dividing powers with the same base—after they are introduced in order. Students can finally combine these techniques to simplify more difficult phrases after practicing them.
Activity 1
Have students evaluate \(3^2\). (9) Have students evaluate \(3^5\). (243) Then, write on the board:
\(3^2\) × \(3^5\) = \(3^?\)
“What is the result of \(3^2\) times \(3^5\)?” (2,187) “Is 2,187 a power of 3? Could it be expressed as 3 to a certain exponent?” Allow students to solve it or work it out. They should draw the conclusion that 2,187 = \(3^7\). Once they have arrived at this conclusion, substitute a 7 for the question mark in the previous equation. Then, write:
\(2^3\) × \(2^7\) = \(2^?\)
“What is the result of \(2^3\) times \(2^7\)?” (1,024) “Is 1,024 a power of 2? Could it be expressed as 2 to a certain exponent?" Allow students some time to think about it and work on it. Students will learn the general rule for adding exponents at different times, so ask them to refrain from giving away their solutions so that other students can figure it out and find the pattern on their own. When students determine that \(2^3\) × \(2^7\) = \(2^{10}\), substitute the question mark with a 10.
"Is there a connection between the exponents on the equal sign's left and right sides?" (One is the total of the other two.)
Ask students a variety of questions like: “\(4^5\) times \(4^8\) equals?” (413). Once every student understands the rule for exponents, it should only take a few questions.
"The base, or the number we're raising to the exponent, must be the same when applying this rule." Provide a few counterexamples to demonstrate this, such as:
\(5^3\) × \(2^7\)
Describe how this is an instance where the rule does not apply because the numbers 5 and 2 are not the same.
For instance, ask students how they arrived at the number \(4^3\). It should be noted that they can locate it by displaying the outcome as 4 × 4 × 4. Similarly, ask them how they could find the result of \(4^5\). (4 × 4 × 4 × 4 × 4) “So, we can write \(4^3\) × \(4^5\) as (4 × 4 × 4) × (4 × 4 × 4 × 4 × 4), which shares the result with \(4^8\).”
Ask students to finish the Multiplying Powers worksheet (M-8-4-1_Multiplying Powers and KEY).
Activity 2
"When multiplying powers with the same base, we found that our rule was that we should add the exponents together. When we divide powers with the same base, what should happen to the exponents? Does anybody have any ideas?" At some point, students will likely guess and subtract them from each other.
“Let’s see. Let’s try \(6^5\) ÷ \(6^2\). What is the result of \(6^5\)?” (7,776) “And what about \(6^2\)?” (36) “And what is the value of 7,776 ÷ 36?” (216) “So, following our hypothesis, we ought to subtract the exponents, and we should come to the result of \(6^3\). Is \(6^3\) equal to 216?” (Yes.) "So it looks like subtraction is a good rule. Exponents should be subtracted when dividing powers with the same base."
Write \(7^9 \over 7^3\) on the board. "Which order should the exponents be subtracted in this problem? Is it better to go 3 – 9 or 9 – 3?" Help students comprehend that we subtract the exponent of the denominator from the exponent of the numerator or the exponent of the divisor from the exponent of the dividend. Make sure to clarify that you can have negative exponents; therefore, you can't use the assumption that it must be 9–3 if students claim that it must be, or else you get a negative number or you can't have negative exponents.
“Now, let’s see how we can perform a problem like \(7^9 \over 7^3\). We can write it as: \(7×7×7×7×7×7×7×7×7 \over 7×7×7\)
"After that, we can remove a few 7s from both the denominator and the numerator. How many 7s can we cancel?" (Three). "Well, those will be cancelled out." Eliminate three 7s from the denominator and the numerator. "What remains within the numerator?" (7 × 7 × 7 × 7 × 7 × 7 or \(7^6\) ). "In the denominator, what remains?" Help students understand that the denominator in this case is still 1, even if we've canceled everything out.
“So we have \(7^6\) divided by 1, but that’s just \(7^6\). It looks like algebra supports our rule.”
Ask students to finish the Dividing Powers worksheet (M-8-4-1_Dividing Powers and KEY).
Activity 3
"We're going to put everything together now and examine expressions that divide and multiply powers that have the same base."
Spread out 14 copies of the Powers in Expressions packet (M-8-4-1_Powers in Expressions) among 14 different students. Note: You can use fewer pages or make more pages if you need to. There isn't a hard and fast rule for how many pages you should use, as long as there are enough × or ÷ pages.
Assign students to groups according to whether they have a paper with an operation (either × or ÷) or a number (a power of 7). Students with operations and students with numbers should each form a single-file line. Subsequently, assign a number to each student and have them come up and demonstrate it. Then, assign a number to each student who has an operation, another number, etc., and so on.
They should stand next to one another to form an expression like \(7^8\) × \(7^{-4}\). Have students respond with the simplified value of the expression (expressed as a power of 7, in this case, \(7^4\)).
You can extend this exercise by having students from the head of each line come forward and continue the expression, provided that it alternates between a number and an operation. Simple expressions, or those that only involve one operation, can be continued at this point. A longer version of this could be:
\(7^8\) × \(7^5\) ÷ \(7^2\) × \(7^{-4}\)
Remind the class that the left-to-right method is used for both multiplication and division. For instance, the following expression can be made simpler:
\(7^{13}\) ÷ \(7^2\) × \(7^{-4}\)
\(7^{11}\)× \(7^{-4}\)
\(7^7\)
Students can engage in a variety of interactions with those creating the expressions, depending on the class:
For future assessment, students can write down their responses; in such a scenario, you should also record the replies.
students can just answer out of turn verbally.
students can raise their hands to answer verbally.
Students can form teams, and each team needs to raise its hand and give a spoken response.
Students who possess numbers and operations can switch places with those who don't once the exercise has been going on for a while. (Therefore, to ensure that every student has an equal opportunity at both portions of the activity, it might be appropriate to assign around half of the class to numbers and operations.)
Extension:
You can change the lesson to fit the needs of the students all year by using these strategies.
Routine: You may use these concepts all year long. These kinds of simplification tasks can be utilized as problems of the day when students respond to an opening question at the start of class regarding the subject because they are quite quick to develop and solve. They may be used in this way with effectiveness and don't take too much time away from other, new concepts because of their simplicity.
Small Group: Give this exercise to students who could use a little more experience. Divide the class into groups using the smaller copies of the Powers in Expressions packet. Distribute copies of the Powers in Expressions packet to every student, and ask one to provide a response, like \(7^{22}\) Using their cards, other students should come up with an expression whose simplified value equals the student's response. (Students can also be given cards with parenthesis on them for added variation and challenge.)
Expansion: Use this adjustment with students who are willing to go above and beyond the requirements of the standard. More rational exponents, negative exponents, and powers using variables as bases can all be added to the lessons covered in this module. (For instance, \(x^5\) × \(x^{11}\) = \(x^?\).) Students can build upon and apply these skills in a variety of ways as they work further with simplifying expressions and solving equations and inequalities.
