# Lesson 13

Cube Roots

Let’s compare cube roots.

### 13.1: True or False: Cubed

Decide if each statement is true or false.

$$\left( \sqrt[3]{5} \right)^3=5$$

$$\left(\sqrt[3]{27}\right)^3 = 3$$

$$7 = \left(\sqrt[3]{7}\right)^3$$

$$\left(\sqrt[3]{10}\right)^3 = 1,\!000$$

$$\left(\sqrt[3]{64}\right) = 2^3$$

### 13.2: Cube Root Values

What two whole numbers does each cube root lie between? Be prepared to explain your reasoning.

1. $$\sqrt[3]{5}$$
2. $$\sqrt[3]{23}$$
3. $$\sqrt[3]{81}$$
4. $$\sqrt[3]{999}$$

### 13.3: Solutions on a Number Line

The numbers $$x$$, $$y$$, and $$z$$ are positive, and:

$$\displaystyle x^3= 5$$

$$\displaystyle y^3= 27$$

$$\displaystyle z^3= 700$$

1. Plot $$x$$, $$y$$, and $$z$$ on the number line. Be prepared to share your reasoning with the class.
2. Plot $$\text- \sqrt[3]{2}$$ on the number line.

Diego knows that $$8^2=64$$ and that $$4^3=64$$. He says that this means the following are all true:

• $$\sqrt{64}=8$$
• $$\sqrt[3]{64}=4$$
• $$\sqrt{\text -64}=\text-8$$
• $$\sqrt[3]{\text -64}=\text -4$$

Is he correct? Explain how you know.

### Summary

Remember that square roots of whole numbers are defined as side lengths of squares. For example, $$\sqrt{17}$$ is the side length of a square whose area is 17. We define cube roots similarly, but using cubes instead of squares. The number $$\sqrt[3]{17}$$, pronounced “the cube root of 17,” is the edge length of a cube which has a volume of 17.

We can approximate the values of cube roots by observing the whole numbers around it and remembering the relationship between cube roots and cubes. For example, $$\sqrt[3]{20}$$ is between 2 and 3 since $$2^3=8$$ and $$3^3=27$$, and 20 is between 8 and 27. Similarly, since 100 is between $$4^3$$ and $$5^3$$, we know $$\sqrt[3]{100}$$ is between 4 and 5. Many calculators have a cube root function which can be used to approximate the value of a cube root more precisely. Using our numbers from before, a calculator will show that $$\sqrt[3]{20} \approx 2.7144$$ and that $$\sqrt[3]{100} \approx 4.6416$$.

Also like square roots, most cube roots of whole numbers are irrational. The only time the cube root of a number is a whole number is when the original number is a perfect cube.

### Glossary Entries

• cube root

The cube root of a number $$n$$ is the number whose cube is $$n$$. It is also the edge length of a cube with a volume of $$n$$. We write the cube root of $$n$$ as $$\sqrt[3]{n}$$.

For example, the cube root of 64, written as $$\sqrt[3]{64}$$, is 4 because $$4^3$$ is 64. $$\sqrt[3]{64}$$ is also the edge length of a cube that has a volume of 64.