• Students may be assessed on their completion of the Multiplying Powers worksheet.
• Students’ understanding may be evaluated based on their completion of the Dividing Powers worksheet .
• Student performance and teacher observation during Activity 3 using Powers in Expressions (M-8-4-1_Powers in Expressions.docx) will aid in determining level of student mastery.

Activity 1

Ask students to evaluate 3². (9) Ask students to evaluate 3⁵. (243) Then, write on the board:

3² × 3⁵ = 3^?

“What is the value of 3² times 3⁵?” (2,187) “Is 2,187 a power of 3? Can it be written as 3 to some exponent?” Give students time to experiment or work it out. They should arrive at the conclusion that 2,187 = 3⁷. After they draw this conclusion, replace the question mark in the equation above with a 7. Then, write:

2³ × 2⁷ = 2^?

“What is the value of 2³ times 2⁷?” (1,024) “Is 1,024 a power of 2? Can it be written as 2 to some exponent?” Give students some time to think about it and work on it. Students will discover the general rule for adding exponents at different times, so ask students to not shout out their answers to allow other students to work it out and discover the pattern on their own. Once students determine that 2³ × 2⁷ = 2¹⁰, replace the question mark with a 10.

“Is there a relationship between the exponents on the left-hand side of the equal sign and the right-hand side of the equal sign?” (One is a sum of the other two.)

Ask students a variety of questions such as: “4⁵ times 4⁸ equals?” (4¹³). It should only take a few questions before every student has mastered the rule for exponents.

“It’s important when you use this rule that the base, the number we’re raising to the exponent, is the same.” Illustrate this with a couple of counterexamples such as:

5³ × 2⁷

Explain that this is a case in which the rule does not apply because 5 and 2 are not the same number.

Ask students how, for example, they could find the value of 4³. They should note that they can find it by representing the result as 4 × 4 × 4. Likewise, ask them how they could find the value of 4⁵. (4 × 4 × 4 × 4 × 4) “So, we can write 4³ × 4⁵ as (4 × 4 × 4) × (4 × 4 × 4 × 4 × 4), which is the same as 4⁸.”

Have students complete the Multiplying Powers worksheet (M-8-4-1_Multiplying Powers and KEY.docx).

Activity 2

“When we multiplied powers with the same base, we found that our rule was that we should add the exponents together. Does anyone have any guesses as to what we should do to the exponents when we divide powers with the same base?” Students will most likely guess, at some point, to subtract them from one another.

“Let’s take a look. Let’s try 6⁵ ÷ 6². What is 6⁵?” (7,776) “And what is 6²?” (36) “And what is 7,776 ÷ 36?” (216) “So, according to our hypothesis, we should subtract the exponents and we should end up with 6³. Is 6³ equal to 216?” (Yes.) “So it looks like subtraction is a good rule. When dividing powers with the same base, subtract the exponents.”

Write  on the board. “For this problem, in which order should we subtract the exponents? Should we do 9 – 3 or 3 – 9?” Guide students toward understanding that we subtract the exponent of the divisor from the exponent of the dividend, or the exponent of the denominator from the exponent of the numerator. If students say that it must be 9 – 3 because otherwise you get a negative number and/or you can’t have negative exponents, make sure to point out that you can have negative exponents so you can’t use that as a general rule.

“Now, let’s look at how we can represent a problem like . We can write it as: .

“Then, we can cancel some 7s from the numerator and denominator. How many 7s can we cancel?” (three) “So, we’ll cancel those out.” Cross out three 7s from both the numerator and denominator. “What’s left in the numerator?” (7 × 7 × 7 × 7 × 7 × 7 or 7⁶). “What’s left in the denominator?” Here, make students recognize that even though we’ve canceled everything out, the denominator is still 1.

“So we have 7⁶ divided by 1, but that’s just 7⁶. So it looks like the algebra supports our rule.”

Have students complete the Dividing Powers worksheet (M-8-4-1_Dividing Powers and KEY.docx).

Activity 3

“Now, we’re going to combine it all together and look at expressions that multiply and divide powers with the same base.”

Give 14 students each a single page from the Powers in Expressions packet (M-8-4-1_Powers in Expressions.docx). Note: Use fewer pages or create additional pages as necessary; there’s really no limit on the number of pages to use as long as there are sufficient × and ÷ pages.

Separate students based on whether they have a page with a number (a power of 7) or an operation (either × or ÷). Have each group (students with numbers and students with operations) form a single-file line. Then, have the first student with a number come up and show the number, followed by the first student with an operation, followed by another student with a number etc. They should stand next to one another to form an expression like 7⁸ × 7⁻⁴. Have students respond with the simplified value of the expression (expressed as a power of 7, in this case, 7⁴).

This activity can be continued with simple expressions (those involving a single operation) but can be expanded by continuing to have students from the front of each line come up and continue the expression, as long as the expression alternates between number and operation. A sample longer expression might be:

7⁸ × 7⁵ ÷ 7² × 7⁻⁴

Remind students that multiplication and division are performed from left to right. The above expression, for example, can be simplified:

7¹³ ÷ 7² × 7⁻⁴

7¹¹ × 7⁻⁴

7⁷

Depending on the class, students can interact with those making the expressions in several different ways:

• Students can write down their answers for assessment later (in which case you should record the answers as well).
• Students can simply respond verbally out of turn.
• Students can respond verbally by raising their hand.
• Teams can be formed, and students from each team must raise their hands and respond verbally.

After the activity has gone on awhile, students with numbers and operations can exchange places with those that don’t have numbers and operations. (Therefore, having about half the class with numbers and operations might be suitable so every student can have a chance at both parts of the activity.)

Extension:

Use the following strategies to tailor the lesson to meet the needs of  students throughout the year.

• Routine: These concepts can be used throughout the year. Because simplification problems of this type are relatively quick (both to create and solve), they can be used as problems of the day in which students answer a question about this topic at the beginning of class. Due to the simplicity, they can be used effectively in this manner without taking too much time away from other, new concepts.
• Small Group: Use this activity with students who might benefit from additional practice. Using smaller versions of the Powers in Expressions packet, have students split up into groups. Give each student a copy of Powers in Expressions packet and have one student give an answer, such as 7²². Other students should use their cards to create an expression that, when simplified, has a value equal to the answer the student provided. (For more variety and difficulty, students can be given cards with parentheses on them as well.)
• Expansion: Use this modification for students who are prepared to go beyond the requirements of the standard. The concepts in this lesson can be expanded to include more negative exponents, rational exponents, and powers with variables as bases. (For example, x⁵ × x¹¹ = x^?.) As students work more with simplifying expressions and solving equations and inequalities, these skills can be expanded upon and utilized in different ways.