If students have trouble getting started, consider asking:

• “How did you decide which three equations went together in the Warm-up?”

• “How could writing out the repeated factors, for example, writing as , help you find the value of the variable?”

The purpose of this discussion is to review some useful exponent rules. Invite students to share their solutions and reasoning.

Each of these exponent rules is an equation in which the expression on the left uses two different exponents and the expression on the right uses only a single exponent. For each exponent rule, display only the left side, and ask students what the exponent rule is for writing an equivalent expression that uses only a single exponent.

Keep the exponent rules displayed until the end of the lesson.

For division to make sense with exponents, negative integer exponents must represent repeated factors that are the reciprocal of the base, and an exponent of 0 must mean the expression has a value of 1. The key idea is to frame the discussion of 0 and negative integer exponents in terms of comparing numerator and denominator in a division of powers of the same base. Here are some questions for discussion:

• “How can you tell that is greater than 1 without doing any computation?” (There are more factors that are 2 in the numerator, so the numerator is greater than the denominator.)
• “How can you tell that is less than 1 without doing any computation?” (There are fewer factors that are 2 in the numerator, so the numerator is less than the denominator.)
• “How can you tell that is equal to 1 without doing any computation?” (The numerator and denominator are identical, so the expression has a value of 1.)
• “If , what can you say about and ?” ( is 5 more than .)
• “Looking at , what does the exponent 0 mean?” (In a fraction with powers of 2 in the numerator and denominator, the numerator and denominator have the same number of factors, so the value of the fraction is 1.)
• “What changes if the base is 5 or 3 or a non-zero value of  instead of 2? What stays the same?” (The bases are different numbers, but the structure of the division is exactly the same. Negative exponents still correspond to more factors in the denominator, and an exponent of 0 still corresponds to a value of 1.)

Ask groups to share which expressions they thought were equivalent and to explain their reasoning. Invite the selected students to share about the difference between and or, if no groups discussed the idea, ask the class to discuss it now.

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone use a different way to decide that expressions matched?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”