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nth Root

The "nth Root" of a given value, used n times in a multiplication, equals the given value

" nth "

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

Instead of talking about the "4th", "16th", etc, if we want to talk generally we say the "nth".

The nth Root

square root a Just like the square root is what is used two times in a multiplication to get the original value ...
cube root a ... and the cube root is what is used three times in a multiplication to get the original value ...
nth root a ... the nth root is what is used n times in a multiplication to get 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: nth root 625 is 5 , what is "n"?

Answer: 5 × 5 × 5 × 5 = 625, so n=4 (ie 5 is used 4 times in a multiplication)

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

Example: If n is odd then nth root a^n

Why "Root" ... ?

tree root

When you see "root" think

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

In the case of √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

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

nth root ab
(assuming a and b both ≥ 0)

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

Example: cube root 128

It also works for division:

nth root a divide b
(a≥0 and b>0)
(b cannot be zero or you would be dividing by zero)

Example: root3 1 divide 64

Addition and Subtraction

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

no! cannot distribute add or subtract under nth root no!

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

arrow If   a to the nth equals b   then   a=nth root b   (b ≥ 0) Example:5 power 4

nth root of a-to-the-nth-power

When a value has an exponent of n and you then take the nth root, you get the value back again (or sometimes the absolute value of it):

arrow nth root a^n (for a ≥ 0)
nth root a^n (for any value of a, and n is odd)
nth root a^n(for any value of a, and n is even)
      (Note: |a| means the absolute value of a)
root examples


nth root of a-to-the-mth-power

Now let's see what happens when the exponent and root are different values (m and n).

arrow a Example:cube root of 27 squared

So ... you can move the exponent "out from under" the nth root, which may sometimes be helpful.

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

arrow a Example:cube root of 4 to the 6th

That is because the nth root is the same as an exponent of (1/n):

arrow a Example: 2½ = √2 (the square root of 2)

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