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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 ${3}^{2}=9$, we say that the square root of $9$, written as $\sqrt{9}$, is $3$.
${3}^{2}=9\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}3=\sqrt{9}$
Similarly, to find the cube root of a number $x$, we look for a number whose cube is $x$. For example, since ${2}^{3}=8$, we say that the cube root of $8$, written as $\sqrt[3]{8}$, is $2$.
${2}^{3}=8\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}2=\sqrt[3]{8}$

${4}^{\text{th}}$‍  roots

Let's continue that pattern! To find the ${4}^{\text{th}}$ root of a number $x$, we look for a number which, raised to the ${4}^{\text{th}}$ power, equals $x$. For example, since ${3}^{4}=81$, we say that the ${4}^{\text{th}}$ root of $81$, written as $\sqrt[4]{81}$, is $3$.
${3}^{4}=81\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}3=\sqrt[4]{81}$

Let's practice finding some ${4}^{\text{th}}$‍  roots

Problem 1.1
$\sqrt[4]{16}=$

${5}^{\text{th}}$‍  roots

And our journey continues! To find the ${5}^{\text{th}}$ root of a number $x$, we look for a number which, raised to the ${5}^{\text{th}}$ power, equals $x$. For example, since ${2}^{5}=32$, we say that the ${5}^{\text{th}}$ root of $32$, written as $\sqrt[5]{32}$, is $2$.
${2}^{5}=32\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}2=\sqrt[5]{32}$

Let's practice finding some ${5}^{\text{th}}$‍  roots

Problem 2.1
$\sqrt[5]{243}=$

Other higher order roots

We can continue this way and define ${6}^{\text{th}}$ roots, ${7}^{\text{th}}$ roots etc. For example, ${3}^{6}=729$, so the ${6}^{\text{th}}$ root of $729$, written as $\sqrt[6]{729}$, is $3$.

Let's do some more problems with higher order roots

Problem 3.1
$\sqrt[7]{128}=$

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.
• So tough but finally figured it out
• Congratulations. Is this school required for you? If so, I feel your pain.😬
• how do you do this?
• The basic logic behind roots is that you take the number and do the inverse of the exponents to arrive at your answer.

For Example:-

Solve.

cube root of 343

if you have memorized the cube roots you know it is 7, but lets look at the algebraic steps to complete this question.

343 can be further divided to - 49 x 7

49 can be divided down to - 7 x 7

So, if you count up the '7's you see, you will see that there are three.

So the answer would be 3.

It's that easy!

(unless if you don't get it, then I feel stupid saying that last part.)