# 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.

- \(\sqrt[3]{5}\)
- \(\sqrt[3]{23}\)
- \(\sqrt[3]{81}\)
- \(\sqrt[3]{999}\)

### 13.3: Solutions on a Number Line

The numbers \(x\), \(y\), and \(z\) are positive, and:

\(\displaystyle x^3= 5\)

\(\displaystyle y^3= 27\)

\(\displaystyle z^3= 700\)

- Plot \(x\), \(y\), and \(z\) on the number line. Be prepared to share your reasoning with the class.
- 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.