nth Root

The "nth Root" used n times in a multiplication gives the original value

" nth ? "

1st, 2nd, 3rd, 4th, 5th, ... nth ...

Instead of talking about the "4th", "16th", etc, we can just say the "nth ".

The nth Root

2   a × a = a   The square root used two times in a multiplication gives the original value.
3   3a × 3a × 3a = a   The cube root used three times in a multiplication gives the original value.

n   na × na × ... × na = a
(n of them)
  The nth root used n times in a multiplication gives the original value.

So it is the general way of talking about roots
(so it could be 2nd, or 9th, or 324th, or whatever)

The nth Root Symbol

  nth root symbol

This is the special symbol that means "nth root", it is the "radical" symbol (used for square roots) with a little n to mean nth root.

Using it

We could use the nth root in a question like this:

Question: What is "n" in this equation?

n625 = 5

Answer: I just happen to know that 625 = 54 , so the 4th root of 625 must be 5:

4625 = 5

Or we could use "n" because we want to say general things:

Example: When n is odd then   nan = a   (we talk about this later).

Why "Root" ... ?

tree root

When you see "root" think

"I know the tree, but what is the root that produced it? "

Example: in √9 = 3 the "tree" is 9 , and the root is 3 .


Now we know what an nth root is, let us look at some properties:

Multiplication and Division

We can "pull apart" multiplications under the root sign like this:

nab = na × nb
(Note: if n is even then a and b must both be ≥ 0)

This can help us simplify equations in algebra, and also make some calculations easier:


3128 = 364×2 = 364 × 32 = 432

So the cube root of 128 simplifies to 4 times the cube root of 2.

It also works for division:

na/b = na / nb
(a≥0 and b>0)
Note that b cannot be zero, as we can't divide by zero


31/64 = 31 / 364 = 1/4

So the cube root of 1/64 simplifies to just one quarter.

Addition and Subtraction

But we cannot do that kind of thing for additions or subtractions!

no!   na + b na + nb

no!   na − b nanb

no!   nan + bn a + b

Example: Pythagoras' Theorem says

Right angled triangle   a2 + b2 = c2

So we calculate c like this:

c = a2 + b2

Which is not the same as c = a + b , right?

It is an easy trap to fall into, so beware.

It also means that, unfortunately, additions and subtractions can be hard to deal with when under a root sign.


Exponents vs Roots

An exponent on one side of "=" can be turned into a root on the other side of "=":

If  an = b  then  a = nb

Note: when n is even then b must be ≥ 0


54 = 625  so  5 = 4625


nth Root of a-to-the-nth-Power

When a value has an exponent of n and we take the nth root we get the value back again ...

... when a is positive (or zero):

  nth root a^n   (when a ≥ 0 )

Example: root examples

... or when the exponent is odd :

  nth root a^n   (when n is odd )

Example:root examples

... but when a is negative and the exponent is even we get this:

Square root of square

Did you see that −3 became +3 ?

... so we must do this:
  nth root a^n = abs(a)   (when a < 0 and n is even )

The |a| means the absolute value of a, in other words any negative becomes a positive.

Example:4th root example

So that is something to be careful of! Read more at Exponents of Negative Numbers

Here it is in a little table:

  n is odd n is even
a ≥ 0 nth root a^n nth root a^n
a < 0 nth root a^n nth root a^n = abs(a)


nth Root of a-to-the-mth-Power

What happens when the exponent and root are different values (m and n)?

Well, we are allowed to change the order like this:

nam = (na )m

So this:    nth root of (a to the power m)
becomes  (nth root of a) to the power m


3272 = (327 )2
= 32
= 9

Easier than squaring 27 then taking a cube root, right?

But there is an even more powerful method ... we can combine the exponent and root to make a new exponent, like this:

nam = amn

The new exponent is the fraction mn which may be easier to solve.


346 = 463
= 42
= 16

This works because the nth root is the same as an exponent of (1/n)

na = a1n


29 = 912 = 3

You might like to read about Fractional Exponents to find out why!


318, 2055, 319, 317, 1087, 2056, 1088, 2057, 3159, 3160