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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 3, squared, equals, 9, we say that the square root of 9, written as square root of, 9, end square root, is 3.
3, squared, equals, 9, \Longleftrightarrow, 3, equals, square root of, 9, end square root
Similarly, to find the cube root of a number x, we look for a number whose cube is x. For example, since 2, cubed, equals, 8, we say that the cube root of 8, written as cube root of, 8, end cube root, is 2.
2, cubed, equals, 8, \Longleftrightarrow, 2, equals, cube root of, 8, end cube root

4, start superscript, start text, t, h, end text, end superscript roots

Let's continue that pattern! To find the 4, start superscript, start text, t, h, end text, end superscript root of a number x, we look for a number which, raised to the 4, start superscript, start text, t, h, end text, end superscript power, equals x. For example, since 3, start superscript, 4, end superscript, equals, 81, we say that the 4, start superscript, start text, t, h, end text, end superscript root of 81, written as root, start index, 4, end index, is 3.
3, start superscript, 4, end superscript, equals, 81, \Longleftrightarrow, 3, equals, root, start index, 4, end index

Let's practice finding some 4, start superscript, start text, t, h, end text, end superscript roots

Problem 1.1
root, start index, 4, end index, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

5, start superscript, start text, t, h, end text, end superscript roots

And our journey continues! To find the 5, start superscript, start text, t, h, end text, end superscript root of a number x, we look for a number which, raised to the 5, start superscript, start text, t, h, end text, end superscript power, equals x. For example, since 2, start superscript, 5, end superscript, equals, 32, we say that the 5, start superscript, start text, t, h, end text, end superscript root of 32, written as root, start index, 5, end index, is 2.
2, start superscript, 5, end superscript, equals, 32, \Longleftrightarrow, 2, equals, root, start index, 5, end index

Let's practice finding some 5, start superscript, start text, t, h, end text, end superscript roots

Problem 2.1
root, start index, 5, end index, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Other higher order roots

We can continue this way and define 6, start superscript, start text, t, h, end text, end superscript roots, 7, start superscript, start text, t, h, end text, end superscript roots etc. For example, 3, start superscript, 6, end superscript, equals, 729, so the 6, start superscript, start text, t, h, end text, end superscript root of 729, written as root, start index, 6, end index, is 3.

Let's do some more problems with higher order roots

Problem 3.1
root, start index, 7, end index, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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