# Lesson 3

Exponents That Are Unit Fractions

### Lesson Narrative

In this lesson, students connect roots to exponents that are unit fractions. In particular, students extend exponent rules to cases in which exponents are $$\frac12$$ and $$\frac13$$ to find that $$b^{\frac12}=\sqrt{b}$$ and $$b^{\frac13}=\sqrt[3]{b}$$. Students will occasionally encounter exponents of $$\tfrac14$$ and $$\tfrac15$$ throughout the unit, but square and cube roots will be the main focus. From this foundation, students can generalize to exponents of the form $$\tfrac1n$$.

Students make sense of expressions like $$3^{\frac12}$$ 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.

### Learning Goals

Teacher Facing

• Explain how to match expressions involving exponents with equivalent expressions.
• Justify the equivalence of $b^{1/n}$ and $\sqrt[n]{b}$ using the properties of exponents.

### Student Facing

• Let’s explore exponents like $$\frac12$$ and $$\frac13$$.

### Student Facing

• I can write square and cube roots as exponents.

Building On