This Math Talk focuses on rational exponents. It encourages students to think about fractional and negative exponents and to rely on the properties of exponents to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students estimate values from an exponential graph.

To make sense of a negative fractional exponent, students need to look for and make use of structure (MP7).

The goal of this discussion is to review students’ strategies for understanding fractional and negative exponents.

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?”

If this strategy is not mentioned, consider asking students how they would use  or to evaluate the last expression. If the results of the pre-assessment showed that students need extra practice with exponent rules, highlight any student strategies that leverage these rules and carefully record student thinking for all to see.

Students practice identifying expressions involving rational exponents with equivalent radical expressions. In this activity, students are asked to use their understanding of roots and exponents to answer the problems, so technology is not an appropriate tool.

Students build on their work in the previous activity to practice their understanding of rational exponents by finding expressions that have the same value.