## 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 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

## Properties

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:

### Example:

=
=

×
=
4

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

=

=
1/4

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

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

### Example:

54 = 625  so  5 =

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

### Example:

=

= 32
= 9

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

=

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

### Example:

=
= 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