Cube Root
In short: A cube root of a number is the value that gives the number back when it is multiplied by itself three times, written with the radical symbol and a small 3.
8th GradeNumbers and Operations8.EE.A.2
What is Cube Root?
A cube root undoes cubing. Because \(7 \times 7 \times 7 = 343\), the cube root of 343 is 7, written \(\sqrt[3]{343} = 7\). It answers the question: what edge length gives this volume for a cube?
Unlike square roots, each real number has exactly one real cube root, and its sign matches the number: \(\sqrt[3]{-64} = -4\) because \((-4)^3 = -64\). Cubes of whole numbers, 1, 8, 27, 64, 125, 216, are called perfect cubes.
How to spot it
- The volume of a cube is given and the length of an edge is requested.
- An equation has the form x cubed equals a number, and x must be found.
- The radical sign carries a small 3 in the crook of the root.
- A negative number sits under the root and the question asks whether a real answer exists.
The method
- Ask which number multiplied by itself three times equals the radicand; test perfect cubes such as 27, 64, 125, 216.
- Keep the sign: a negative radicand gives a negative cube root.
- If it is not a perfect cube, bracket it between the cube roots of the nearest perfect cubes and estimate.
- Check by cubing your answer.
Cube Root: solved example
Problem
A cube-shaped shipping crate holds 343 cubic feet. What is the length of one edge? And what is \(\sqrt[3]{-64}\)?
Solution
The edge \(e\) satisfies \(e^3 = 343\), and \(7^3 = 343\), so \(e = 7\) feet.
For the second value, \((-4)^3 = -64\), so \(\sqrt[3]{-64} = -4\). Answers: 7 feet and \(-4\).
Try it
Question 1
Evaluate \(\sqrt[3]{216}\) and \(\sqrt[3]{\frac{8}{27}}\).
Show the answer
\(6^3 = 216\), so the first is 6. For the second, \(\frac{2}{3}\) cubed is \(\frac{8}{27}\), so the root is \(\frac{2}{3}\).
Question 2
A cubic planter holds 1,728 cubic inches of soil. How long is its edge?
Show the answer
\(12^3 = 1{,}728\), so the edge is 12 inches.
Cube Root vs. Square Root
A square root asks which number times itself gives the target and is tied to area; a cube root asks which number used three times as a factor gives the target and is tied to volume. Negative numbers have a real value here but no real square root, and a positive number has two square roots but one cube root.
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