nth Root

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

Instead of talking about the «4th», «16th», etc, we can just say the «».

The nth Root

  • The «2nd» root is the square root
  • The «3rd» root is the cube root
  • etc!

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?

= 5

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

= 5

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

Example: When n is odd then  

= a

  (we talk about this later).

Why «Root» … ?


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:



(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:




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

It also works for division:


(a≥0 and b>0)
Note that b cannot be zero, as we can’t divide by zero



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!



a − b

an + bn
a + b

Example: Pythagoras’ Theorem says

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 =

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


54 = 625  so  5 =

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

Square root of square

Did you see that −3 became +3 ?

Here it is in a little table:

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:


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



= 32
= 9

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


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


= 42
= 16

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


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

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