In this lesson, students connect roots to exponents that are unit fractions. In particular, students extend exponent rules to cases in which exponents are \(\begin{array}{l}\frac12\\\end{array}\) and \(\begin{array}{l}\frac13\\\end{array}\) to find that \(b^{\frac12}=\sqrt{b}\) and \(b^{\frac13}=\sqrt[3]{b}\). Students will occasionally encounter exponents of \(\begin{array}{l}\frac14\\\end{array}\) and \(\begin{array}{l}\frac15\\\end{array}\) throughout the unit, but square and cube roots will be the main focus. From this foundation, students can generalize to exponents of the form \(\begin{array}{l}\frac1n\\\end{array}\).
Students make sense of expressions like \(\begin{array}{l}3^{\frac12}\\\end{array}\) using graphs and exponent rules to find that they are roots (MP1).
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview
 3.1 Warmup: Sometimes It’s Squared and Sometimes It’s Cubed (5 minutes)
 3.2 Activity: To the...Half? (15 minutes)
 3.3 Activity: Fraction of What, Exactly? (5 minutes)

3.4 Activity: Exponents and Radicals (10 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 3.5 Cooldown: Write It Two Ways (5 minutes)
Learning goals:
 Explain how to match expressions involving exponents with equivalent expressions.
 Justify the equivalence of \(\begin{array}{l}b^{1/n}\\\end{array}\)and \(\begin{array}{l}\sqrt[n]b\\\end{array}\) using the properties of exponents.
Learning goals (student facing):
 Let’s explore exponents like\(\begin{array}{l}\frac12\\\end{array}\)and\(\begin{array}{l}\frac13\\\end{array}\).
Learning targets (student facing):
 I can write square and cube roots as exponents.
Standards:
 This lesson builds on the standards:CCSS.8.EE.A.1MS.8.EE.1MO.8.EEI.A.1CCSS.8.EE.A.2MO.8.EEI.A.2aMO.8.EEI.A.2bMS.8.EE.2CCSS.8.EE.A.3MS.8.EE.3MO.8.EEI.A.3
 This lesson builds towards the standards:CCSS.HSNRN.A.1MS.NRN.1MO.A2.NQ.A.1CCSS.HSNRN.A.2MS.NRN.2MO.A1.NQ.A.2
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