This activity develops fluency with exponent rules and allows students to think flexibly about equivalent expressions. 

Monitor for groups who use these different representations when finding equivalent expressions for . Here are types of expressions students may find, ordered from more common to less common:

• An expression that uses repeated multiplication, such as . 
• An expression that raises a power to another power, such as   or .
• An expression that shows an understanding of negative exponents, such as , which is equivalent to  or .
• An expression that uses multiple exponent rules at once, such as .

The purpose of this discussion is for students to see multiple ways to represent the same expression. 

Invite previously selected groups to share their equivalent expression for . Sequence the discussion of the expressions in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions such as these:

• “Is one type of expression more useful than another type of expression?” (Sometimes a shorter expression may be easier to write, but sometimes a longer expression might model a particular situation better.)
• “Do you notice any patterns in the different expressions?” (Answers vary.)
• “How did seeing multiple ways to use the exponent rules enhance your understanding of exponents?” (Answers vary.)

In this activity, students analyze the structure of expressions involving different bases and observe that exponents can be added (or subtracted) only when the powers being multiplied (or divided) have the same base (MP7).

It is expected that students will compute the value of the expressions on the left and right sides of the equation to show that they are not actually equal. The last problem alludes to the rule, which will be explored further in a following lesson.

The goal of this discussion is for students to recognize that the exponent rules work because they capture patterns of repeated multiplication of a single base. Invite students to share their responses and display them for all to see. Consider involving more students in the whole-class discussion with the following questions:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone use the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?

For the final question, ask students whether they think it is a coincidence that the equation is true, or if there is another, more general explanation. It is not necessary for students to generalize the relationship here since it will be addressed more fully in a later lesson.