# nth Root

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

## " nth ? "

**1**st, **2**nd, **3**rd, **4**th, **5**th, ... **n**th ...

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

## The nth Root

- The "2nd" root is the square root
- The "3rd" root is the cube root
- etc!

2 | Just like the square root is used two times in a multiplication to get the original value. |
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3 | And the cube root is used three times in a multiplication to get the original value. |
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... | ... | ... |
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n | The nth root 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

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?

Answer: I just happen to know that **625 = 5 ^{4}**, so the

**4**th root of 625 must be 5:

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

Example: When **n** is odd (we talk about this later).

## Why "Root" ... ?

When you see "root" think
Example: in |

## 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:

(*If n is even, a and b must both be ≥ 0)*

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

Example:

It also works for division:

(*a≥0 and b>0)
(b cannot be zero, as we can't divide by zero)*

Example:

### Addition and Subtraction

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

Example: Pythagoras' Theorem says

a^{2} + b^{2} = c^{2} |

So we can calculate c like this:

c = √(a^{2} + b^{2})

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

If then (when n is even b must be ≥ 0) |

Example:

### 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 |
(when a ≥ 0) |

Example:

... or when the |
(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 have: | (when a < 0 and n is even) |

(Note: **|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

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

Example:

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

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

Example: |

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

Example: 2^{½} = √2 (the square root of 2) |

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