If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Higher order roots

In this article, we will expand the concepts of square roots and cube roots to roots of any order.
If you don't know what square and cube roots are, we recommend that you check out this lesson.

Quick review of square and cube roots

To find the square root of a number x, we look for a number whose square is x. For example, since 32=9, we say that the square root of 9, written as 9, is 3.
32=93=9
Similarly, to find the cube root of a number x, we look for a number whose cube is x. For example, since 23=8, we say that the cube root of 8, written as 83, is 2.
23=82=83

4th roots

Let's continue that pattern! To find the 4th root of a number x, we look for a number which, raised to the 4th power, equals x. For example, since 34=81, we say that the 4th root of 81, written as A814, is 3.
34=813=A814

Let's practice finding some 4th roots

Problem 1.1
A164=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

5th roots

And our journey continues! To find the 5th root of a number x, we look for a number which, raised to the 5th power, equals x. For example, since 25=32, we say that the 5th root of 32, written as A325, is 2.
25=322=A325

Let's practice finding some 5th roots

Problem 2.1
A2435=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Other higher order roots

We can continue this way and define 6th roots, 7th roots etc. For example, 36=729, so the 6th root of 729, written as A7296, is 3.

Let's do some more problems with higher order roots

Problem 3.1
A1287=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want to join the conversation?