Cube Root
Definition of Cube Root
To indicate a cube root, we use the same radical symbol we used for
square roots, but we write a small 3 just above the
on the radical
symbol to indicate a cube root. The small 3 is called the index of the
radical.
For example, the cube root of 64 is written
Definition â€”
Cube Root
The cube root of a real number, a, is written
If b is a real number and b^{3}
= a, then
Example:
because 4^{3}
= 64.
Here are some examples:
because 5^{3}
= 5 Â· 5 Â· 5
= 125
because 1^{3}
= 1 Â· 1 Â· 1
= 1.
=
1 because (1)^{3} = (1)(1)(1) = 1.
because (2)^{3}
= (2)(2)(2) = 8.
Note:
To find the cube root of a number, we
reverse the operation of cubing.
For example, to find
we ask â€œWhat
number cubed is 8?â€.
The answer is 2.
Therefore,
The last two examples show that a cube root, unlike a square root, can
have a negative radicand.
We can use geometry to provide a visual interpretation of a cube root.
For example, suppose a cube has a volume of 125 cubic inches. The length
of each side is the cube root of the volume.
That is,
the length of a side of the cube == 5 in.
A perfect cube is a number that has a rational cube root.
For example, 125 is a perfect cube because 5^{3} = 125.
As we work with cube roots, we will find it useful to recognize perfect
cubes and their cube roots.
Now we will summarize the relationship between cubes and cube roots.
Property â€”
Cubes and Cube Roots
English Cubing and taking a cube root â€œundoâ€ each other.
Algebra If a is a real number, then
Example
Note:
Unlike square roots, there are no
â€œprincipalâ€ cube roots.
The sign of a cube root depends on the
sign of its radicand.
Perfect Squares 
Principal Square Roots 
0^{3} = 0 

1^{3} = 1 

2^{3} = 4 

3^{3} = 9 

4^{3} = 16 

5^{3} = 25 

6^{3} = 36 

7^{3} = 49 

8^{3} = 64 

9^{3} = 81 

10^{3} = 100 



Property â€” Cubes and Cube Roots
English Cubing and taking a cube root â€œundoâ€ each other.
Algebra If a is a real number, then
Example
