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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^{\text{th}}$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^{\text{th}}$4, start superscript, start text, t, h, end text, end superscript roots

Problem 1.1
root, start index, 4, end index, equals

## $5^{\text{th}}$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^{\text{th}}$5, start superscript, start text, t, h, end text, end superscript roots

Problem 2.1
root, start index, 5, end index, equals

## 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

## Want to join the conversation?

• it doesn't show you how to solve for a fraction.
• for a fraction, you just find the roots for both the denominator and the numerator. For example, if you are finding the 4th root of 16/243, you find the 4th root of 16(which is 2) and the 4th root of 243(which is 3) so the answer would be 2/3.
Hope that helped
• What are imaginary numbers
• It's a complex number that is written as i, which means the square root of -1, which is impossible.
• I'm not getting this at all i need help
• um so how come when i take a unit test its always something else instead of what i was taught...
• i honestly think so to... but khan is trying to give you a challenge to expand your mind and DIY! thats why they make the tests harder!
• How do you find a decimal root.
• Do you mean the square root of a decimal such as √.64?
• im a bit confused about the 9th root.
• So basically they phrased the question in a tricky way. When they say: "Pick the correct equality that describes 2 as the ninth root of a number." You need to remember that the any root, whether higher order, cube, or square, is the inverse of its counterpart: exponents. So essentially the real question they are asking you is: 2 to the power 9. All you need to do is solve for this and your answer will coincide with one of the multiple choice.