In the previous lesson, students used radicals to rewrite expressions that had unit fraction exponents. In this lesson, students use radicals to rewrite expressions that have other kinds of fractions for exponents, such as by rewriting \(b^\frac{m}{n}\) as \(\sqrt[n]{b^m}\) or \(\left(\sqrt[n]{b}\right)^m\). They do this by breaking \(b^\frac{m}{n}\) into either \((b^m)^{\frac{1}{n}}\) or \(\left(b^{\frac{1}{n}}\right)^m\). In the last activity of the lesson, students find rough approximations for numbers written this way by sketching the graph of \(y=2^x\) from integer values of \(x\) and estimating the \(y\)coordinates on that continuous curve for various positive rational \(x\)coordinates.
Students reason abstractly and quantitatively when they use exponent rules to reason that two exponential expressions have the same value (MP2).
Lesson overview
 4.1 Warmup: Math Talk: Regrouping Fractions (5 minutes)
 4.2 Activity: You Can Use Any Fraction As an Exponent (10 minutes)

4.3 Activity: Fractional Powers Are Just Numbers (20 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 4.4 Cooldown: Third It (5 minutes)
Learning goals:
 Justify the equivalence of \(b^{m/n}\) and \(\sqrt[n]{b^m}\) using the properties of exponents.
 Use graphs to estimate the value of expressions involving positive rational exponents.
Learning goals (student facing):
 Let’s use roots to write exponents that are fractions.
Learning targets (student facing):
 I can interpret exponents that are fractions.
Required materials:
 Scientific calculators
Standards:
 This lesson builds on the standard:CCSS.4.NF.BMS.4.NF
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.