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:
\(\displaystyle x^3= 5\)
\(\displaystyle y^3= 27\)
\(\displaystyle z^3= 700\)
![A numbre line that shows the integers from negative 3 to 9](https://cms-im.s3.amazonaws.com/G3MZVfcR9KqWSPnJdmP8fu1b?response-content-disposition=inline%3B%20filename%3D%228-8.8.B.5.revised.image.01.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B.5.revised.image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T010709Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ad7dc9e0fa705c8b5af467584e96f50c85c64889ad1083b20a16831dd3cee9f8)
- 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.