# Extend the properties of exponents to rational exponents

## • Extend properties of integer exponents to rational exponents and write expressions with rational exponents as radicals (N-RN.A.1, N-RN.A.2).  • Solve real-world problems in which rational exponents arise (N-RN.A.1).

Students have encountered square roots and cube roots in Grades 6–8. Now that they have the real numbers at their disposal they can contemplate more complicated numerical expressions involving radicals and fractional exponents. In this section they learn the rules for manipulating such expressions. They first review familiar exponent rules and remind themselves how they rewrite exponential expressions, particularly the rule $(x^a)^b = x^{ab}$. They investigate the consequences of extending this rule to rational exponents and see how it implies that $x^{(a/b)}=\sqrt[b]{x^a}$. The section continues with a modeling task using Kepler’s Law and then wraps up with a short reasoning task where students can work with rational exponents in decimal form.

1 Extending the Definitions of Exponents, Variation 2

WHAT: Students are given data for a population of bacteria that is doubling every hour, and asked to think about how to interpolate the data to describe the population every 1/2 hour and then every 1/3 hour (MP.8).

WHY: The purpose of this task is to motivate the definition of $a^\frac{1}{n}$ as $\sqrt[n]{a}$ for a whole number n, using a real-life context. This prepares for the more general definition in the following task.

2 Evaluating Exponential Expressions

WHAT: Students consider some numerical expressions of the form $a^{(m/n)}$ where $a$, $n$, and $m$ are whole numbers and reason through different ways of evaluating them.

WHY: The goal of this task is to develop an understanding for why expressions with rational exponents can be written in terms of radicals by guiding students through equivalent expressions using exponent rules. Introducing rational exponents in this way will help students to hone their ability to reason about complex numerical expressions (MP.2).

3 Kepler's Third Law of Motion

WHAT: In this problem, students explore Kepler’s law for describing the orbit of the planets and solve equations that include a rational exponent.

WHY: A modeling situation is purposefully placed here for students to continue to reason through and wrestle with the meaning of rational exponents as opposed to memorizing a set of procedures and rules (MP.4).

4 Checking a calculation of a decimal exponent

WHAT: This task involves a rational exponent expressed a decimal. Students are asked to rewrite the exponent and reason with inequalities and properties of exponents to prove or disprove statements about the value of the expression.

WHY: The purpose of the task is to connect properties of fractional exponents with ordering of real numbers and further push the students’ thinking about rational exponents and how writing them in one form or another can help solve a problem. The task involves some careful quantitative reasoning (MP.2, MP.6).