This Math Talk focuses on dividing powers of ten and using exponents to rewrite powers of ten. It encourages students to think about exponents and rely on what they know about solving equations to mentally solve problems. The understanding elicited here will be helpful later in this lesson when students will divide expressions using exponents that have the same base.

In this activity, observations about subtracting exponents in the first three equations suggests a method for solving the last equation. Students may be puzzled by the last question. The purpose of including this equation is to conclude that the solution must be -1 because the left side can be rewritten as . The rest of this lesson offers more opportunities to make sense of negative exponents, so it’s not necessary for all students to feel completely comfortable with this idea, yet.

To solve these equations, students need to look for and make use of structure (MP7).

The goal of this discussion is to review students’ strategies for using properties of exponents to solve problems. To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

Draw particular attention to methods that express 100 and 1,000 as powers of 10. For example, if we rewrite as , we can remember our properties of exponents that tell us the denominator must be equal to . The same property of exponents can be used to deduce that the solution to the last equation is -1.

The purpose of this activity is to extend the definition of exponents to include negative exponents. The meaning of an exponent of 0 is also revisited. When students use tables to explore expressions with negative exponents, they are noticing and making use of structure (MP7).