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).

Ask students to share their estimates for and , and display their graphs for all to see. Ask students to share their responses to questions about using the exponent rules and the value of the number raised to the power. The important takeaway is that following the exponent rules means that is another way of writing because, for positive numbers , , and .

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  .

The key takeaway is that extending exponent rules to these fractions means that and . Select students to share their responses for and . Ask students how they could use exponent rules to show that raising a number to the power is equivalent to taking the cube root. (Either , or .)

Conclude the discussion by reminding students that these expressions that involve the symbol are often called radical expressions because the technical name for the symbol is the radix (which is also where the word root comes from). Expressions like and are examples of radical expressions.

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.

If students are not yet sure how to match the negative rational exponents, consider saying:

• “Can you explain how you matched the first few expressions that have integer exponents.”

• “How could finding a pattern help you match to an equivalent expression?”

Begin the discussion by selecting 2–3 groups to share how they matched the expressions with positive exponents. Next, invite previously identified groups to share how they matched the expressions with negative exponents. While students share their thinking, display any exponent rules referred to, such as or . It is important for students to understand that fractional exponents follow all the same exponent rules as integer exponents when the base is positive.

It is not essential to emphasize the restriction to positive bases too heavily to students. If students inquire about the restriction, consider showing them that a value like  is equivalent to ,  which is not possible for any number on the number line. But, if we use an exponent rule, we could do something like , which does not fit with the other understanding.