The purpose of this activity is to elicit strategies and understandings students have for square and cube roots. These understandings help students develop fluency and will be helpful later in this lesson when students will connect square and cube roots to exponents of and . Students might note the negative solutions to the quadratic equations, but it is not necessary to emphasize those solutions at this time if they are not mentioned. Students will explore the symmetry of positive and negative solutions to quadratic equations more deeply in upcoming lessons.

While participating in this activity, students need to be precise in their word choice and use of language (MP6), because they have no algebraic method that they can use to solve and no calculators that could give approximate solutions numerically. They have to be able to recall the definitions of square and cube roots.

Briefly discuss students’ responses. Here are some questions for discussion:

• “How did you find the answers that are not integers?” (I estimated a value for and then cubed it to see how close it was to 19. I used a square root sign to write as .)
• “How could we tell if our answers to these questions make sense?” (We could square or cube them; for example, we know is 2 because .)
• “Could we write or using radicals?” (Yes, because and .)

Key discussion points are that we can write exact answers using radical signs and that we can check the accuracy of our answers by squaring or cubing, as appropriate.

The purpose of this activity is for students to connect numbers raised to the power with square roots. First, they use a graph to estimate the value of such numbers. Then, they extend exponent rules to find that the numbers must be square roots of the base.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

The purpose of this activity is for students to extend what they learned about square roots and exponents of  to cube roots and exponents of  .

In this activity, students match equivalent expressions involving exponents and radicals. This is the first time they see negative rational exponents, so expect a lot of discussion around those expressions. Students will have more chances to interpret negative exponents, so it is not crucial for them to show mastery at this time.

Monitor for groups who reason about the expressions with negative fractional exponents in different ways. For example, some students may see as , while others will think of the expression as . Ask these groups to be ready to share their reasoning during the discussion.