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

 

A numbre line that shows the integers from negative 3 to 9
  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.