Pre-Algebra
Numbers and operations
Cube roots
20 practice questions
0 video lessons
Theory + worked examples
Cube Roots
California Pre-Algebra • Standard 8.EE.2 • Numbers & Operations
Cube Roots is a topic in Numbers & Operations in the California Common Core State Standards. It is aligned to Standard 8.EE.2, which requires students to evaluate cube roots of perfect cubes.
A cube root \(\sqrt[3]{a}\) is the number whose cube is \(a\), and it can be negative.
Theory
The cube root \(\sqrt[3]{a}\) is the number whose cube is \(a\).
Perfect cubes are \(1,8,27,64,125,\ldots\); a cube root can be negative.
What a cube root is.
Squares vs cubes.
Definition:
\[\sqrt[3]{a}=b \iff b^3=a\]
\(\sqrt[3]{-a}=-\sqrt[3]{a}\) — cubes keep the sign.
How to find a cube root
- Ask what number cubed gives the value.
- Recognize perfect cubes.
- Keep the sign (negatives allowed).
- Bracket non-perfect cubes between integers.
Example 1 — Perfect cube
Find \(\sqrt[3]{8}\).
Solution
\(2^3=8\).
| \(\sqrt[3]{8}\) | \(=\) | \(2\) |
Example 2 — Larger cube
Find \(\sqrt[3]{125}\).
Solution
\(5^3=125\).
| \(\sqrt[3]{125}\) | \(=\) | \(5\) |
Example 3 — Negative
Find \(\sqrt[3]{-27}\).
Solution
A cube root can be negative: \((-3)^3=-27\).
| \(\sqrt[3]{-27}\) | \(=\) | \(-3\) |
Example 4 — Estimate
Between which integers is \(\sqrt[3]{30}\)?
Solution
\(27<30<64\), so between \(3\) and \(4\).
| \(3\) | < | \(\sqrt[3]{30}<4\) |
Common pitfalls
Cube roots of negatives exist, unlike square roots.
\(\sqrt[3]{8}=2\), not \(4\) — it is the edge, not a face.
Don't confuse \(\sqrt[3]{a}\) with \(\sqrt{a}\).
Frequently asked questions
What is a cube root?
A number that, cubed, gives the original.
Can a cube root be negative?
Yes.
What is \(\sqrt[3]{27}\)?
\(3\).
What is a perfect cube?
A number whose cube root is a whole number.
More in Numbers and operations