# Lesson 4

Positive Rational Exponents

### Lesson Narrative

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).

### Learning Goals

Teacher Facing

• 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.

### Student Facing

• Let’s use roots to write exponents that are fractions.

### Student Facing

• I can interpret exponents that are fractions.

Building On

Addressing