n^{th} Roots
Now we will study higher order roots, such as cube roots. Like square
roots, these roots can be written using a radical symbol. To indicate the
specific root, a number called the index is written just above the
on
the radical symbol.
For example, the cube root of 8 is written like this:
The index, 3, indicates the radical is a cube root.
The cube root of 8 is 2 because 2^{3} = 8. We write:
Note:
The square root of a number is also called
the 2^{nd} root of the number. The index of a square root is 2, but we
rarely write it. Thus,
The cube root of a number is also called the 3^{rd} root of the number.
In a similar way, we define 4^{th} roots, 5^{th} roots, 6^{th} roots, and so on. For example,
â€¢ The 4^{th} root of 81 is written like this:
. The index is 4.
= 3 because 3^{4} = 81.
â€¢ The 10^{th} root of 1 is written like this:
. The index is 10.
= 1 because 1^{10} = 1.
To indicate an n^{th} root, we use the letter n for the index.
â€¢ If n is odd, then
is always a real number.
For example,
and
are both real numbers:
â€¢ If n is even, then
is a real number only when a
≥ 0.
For example,
, but
is not a real number because
5 Â· 5 ≠
25 and (5) Â· (5)
≠ 25.
In fact, no real number
multiplied by itself will equal 25.
n 
n^{th} root 
symbol 
2 
square root 

3 
cube root 

4 
fourth root 

5 
fifth root 




Example
a. Find
b. Find the 5^{th} root of 243.
Solution
a. Find the prime factorization of 625: 625 = 5 Â· 5
Â· 5 Â· 5 = 5^{4} . Since 5^{4} = 625, and 5 is positive,
= 5.
b. The 5th root of 243 may be written
. Find the prime factorization of
243: 243 = 3 Â· 3
Â· 3 Â· 3
Â· 3 = 3^{5}. Since 3^{5} = 243,
= 3.
