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
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 23 = 8. We write:
The square root of a number is also called
the 2nd 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 3rd root of the number.
In a similar way, we define 4th roots, 5th roots, 6th roots, and so on. For example,
â€¢ The 4th root of 81 is written like this:
. The index is 4.
= 3 because 34 = 81.
â€¢ The 10th root of 1 is written like this:
. The index is 10.
= 1 because 110 = 1.
To indicate an nth root, we use the letter n for the index.
â€¢ If n is odd, then
is always a real number.
are both real numbers:
â€¢ If n is even, then
is a real number only when a
is not a real number because
5 Â· 5 ≠
-25 and (-5) Â· (-5)
In fact, no real number
multiplied by itself will equal -25.
b. Find the 5th root of 243.
a. Find the prime factorization of 625: 625 = 5 Â· 5
Â· 5 Â· 5 = 54 . Since 54 = 625, and 5 is positive,
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 = 35. Since 35 = 243,