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 2³ = 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,

• The 4^th root of 81 is written like this: . The index is 4. = 3 because 3⁴ = 81.

• The 10^th root of 1 is written like this: . The index is 10. = 1 because 1¹⁰ = 1.

To indicate an n^th root, we use the letter n for the index.

• If n is odd, then is always a real number.

For example, and are both real numbers:

• If n is even, then is a real number only when a ≥ 0.

For example, , but is not a real number because 5 · 5 ≠ -25 and (-5) · (-5) ≠ -25.

In fact, no real number multiplied by itself will equal -25.

n	nth 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.

a. Find the prime factorization of 625: 625 = 5 · 5 · 5 · 5 = 5⁴ . Since 5⁴ = 625, and 5 is positive, = 5.

b. The 5th root of 243 may be written .  Find the prime factorization of 243: 243 = 3 · 3 · 3 · 3 · 3 = 3⁵. Since 3⁵ = 243, = 3.