# Lesson 8

Cubes and Cube Roots

### Lesson Narrative

In this lesson, students reason abstractly and quantitatively from the meaning of cube and cube root to solve equations of the form \(x^3 = a\) and \(\sqrt[3]{x} = a\), where \(a\) can be positive or negative (MP2). Students use the graph of \(y=x^3\) to find that all numbers have exactly one cube root.

Note that there are claims like, “All numbers have exactly one cube root,” which would be more precisely stated as “All *real* numbers have exactly one *real* cube root.” However, students don’t know about any numbers other than real numbers, so it does not make sense to make this distinction at this time. In upcoming lessons, students will expand their concept of number to include imaginary and complex numbers.

Some of the activities in this lesson work best when each student has access to devices that can run the Desmos applets because students can attend to the level of precision in making estimates from a graph (MP6).

### Learning Goals

Teacher Facing

- Calculate solutions to equations involving cubes and cube roots and explain the solution method used.
- Compare and contrast the processes of solving equations with square roots and with cube roots.
- Justify using a graph that every number, positive or negative, has exactly one cube root.

### Student Facing

- Let’s compare equations with cubes and cube roots.

### Required Preparation

Devices are required for the digital versions of the activities “Finding Cube Roots with a Graph” and “Cube Root Equations.”

### Learning Targets

### Student Facing

- I can solve equations by cubing or finding cube roots.