Key takeaway: The cube root of a number is the value that, multiplied by itself three times , equals that number, written with the symbol \(\sqrt[3]{\ }\).
Grade 88.EE.A.2Numbers and Operations
Definition of Cube Root
The cube root undoes cubing. Because \(12 \times 12 \times 12 = 1728\), the answer for 1,728 is 12. If a cube-shaped box holds 1,728 cubic inches, each edge measures 12 inches, which explains the name.
Unlike square roots, this operation works for negative numbers too, because three negative factors multiply to a negative result: \((-4)^3 = -64\). Each real number therefore has exactly one real answer. Numbers such as 8, 27, 64, 125 and 1,000 are perfect cubes with whole-number results; the rest are irrational and can be estimated between two perfect cubes.
Formula and notation
The small 3 in the radical is the index:
\[\sqrt[3]{a} = b \ \text{ means } \ b^3 = a\]
For example, \(\sqrt[3]{-125} = -5\), because \((-5)(-5)(-5) = -125\).
Key facts
- Every real number has exactly one real cube root, positive, negative or zero.
- A negative number has a negative result.
- Perfect cubes up to 1,000 are worth memorizing: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
- It appears whenever you know a volume and want an edge length.
Cube Root example
A cube-shaped storage bin holds exactly 1,728 cubic inches. How long is each edge? Then evaluate \(\sqrt[3]{-64}\).
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The volume of a cube is edge cubed, so the edge is \(\sqrt[3]{1728}\).
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Try 12: \(12 \times 12 = 144\) and \(144 \times 12 = 1728\). So the edge is 12 inches.
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For \(-64\): \((-4)(-4)(-4) = -64\), so \(\sqrt[3]{-64} = -4\).
Each edge is 12 inches, and \(\sqrt[3]{-64} = \) \(-4\).
Common mistake
Watch out
Priya computes \(\sqrt[3]{27}\) by dividing: \(27 : 3 = 9\).
Why it is wrong
The little 3 in the symbol means three equal factors, not division by 3. Check: \(9 \times 9 \times 9 = 729\), not 27. The right answer is 3, since \(3 \times 3 \times 3 = 27\).
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Numbers and Operations