# 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 3√a × 3√a × 3√a = a The cube root used three times in a multiplication gives the original value. n n√a × n√a × ... × n√a = 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 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:

## Why "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 .

## 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:

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:

### Example:

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

Example:

31/64 = 31 / 364 = 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! na + b na + nb na − b nanb nan + bn a + b

Example: Pythagoras' Theorem says 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

### Example:

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): (when a ≥ 0 )

Example: ... or when the exponent is odd : (when n is odd )

Example: ... but when a is negative and the exponent is even we get this: Did you see that −3 became +3 ?

 ... so we must do this: (when a < 0 and n is even )

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

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  a < 0  ### 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

### Example:

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.

### Example:

346 = 463
= 42
= 16

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

na = a1n

### Example:

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