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", and so on, we can just say the "nth ".
The nth Root
- The "2nd" root is the square root
- The "3rd" root is the cube root
- and so on!
| 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)
Definition: The nth root of a is the number r that satisfies:
rn = a
And we have r = n√a.
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.
Many Roots?
There can be more than one root. But the "radical symbol" √ means the principal square root.
Example: r2=9
r2=9 has two solutions: r=3 and r=−3
But √9 means the principal square root, which is just 3
Using it
We could use the nth root in a question like this:
Question: What's "n" in this equation?
n√625 = 5
Answer: I just happen to know that 625 = 54 , so the 4th root of 625 must be 5:
4√625 = 5
Or we could use "n" because we want to say general things:
Example: When n is odd then n√an = a (we talk about this later).
Why "Root" ... ?
 |
When you see "root" think "I know the tree, but what's the root that produced it? " Example: in √9 = 3 the "tree" is 9 , and the root is 3 . |
Even Roots of Negative Values
Something to be careful of:
Example: √−4 = 2i, where i is the unit imaginary number.
To stay purely within real numbers don't take even roots of negative values.
Odd Roots of Negative Values
Not a problem! Odd roots of negative numbers work fine with real numbers.
Example: 3√−8 = −2 because (−2) × (−2) × (−2) = −8.
Properties
Now we know what an nth root is, let's look at some properties:
Multiplication and Division
We can "pull apart" multiplications under the root sign like this:
n√ab = n√a × n√b
Except when n is even then a and b can't be negative.
This can help us simplify equations in algebra, and also make some calculations easier:
Example:
3√128 = 3√64×2 = 3√64 × 3√2 = 43√2
So the cube root of 128 simplifies to 4 times the cube root of 2.
It also works for division:
n√a/b = n√a / n√b
(a≥0 and b>0)
Note that b can't be zero, as we can't divide by zero
Example:
3√1/64 = 3√1 / 3√64 = 1/4
So the cube root of 1/64 simplifies to just one quarter.
Rationalizing the Denominator
In mathematics, we prefer not to have a root on the bottom of a fraction. We can move it to the top by a process called Rationalizing the Denominator:
Example: 1√2
Multiply the top and bottom by √2:
1√2 × √2√2 = √22
Addition and Subtraction
But we can't do that kind of thing for additions or subtractions!
n√a + b ≠ n√a + n√b
n√a − b ≠ n√a − n√b
n√an + 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, addition and subtraction can be hard to deal with when under a root sign.
Like Radicals
But we can add or subtract radicals if they are Like Radicals.
Like Radicals have the same index and the same value inside the radical.
Example: What's 2√3 + 5√3 ?
Just like 2x + 5x = 7x, we can add these:
2√3 + 5√3 = 7√3
If the radicals aren't the same, we might be able to simplify them first to see if they become "like".
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 = n√b
Note: when n is even then b must be ≥ 0
Example:
54 = 625 so 5 = 4√625
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 ?
This happens:
... 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's 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:
n√am = (n√a )m
So this: nth root of (a to the power m)
becomes (nth root of a) to the power m
Example:
3√272 = (3√27 )2
= 32
= 9
Easier than squaring 27 then taking a cube root, right?
But there's an even more powerful method ... we can combine the exponent and root to make a new exponent, like this:
n√am = amn
The new exponent is the fraction mn which may be easier to solve.
Example:
3√46 = 463
= 42
= 16
This works because the nth root is the same as an exponent of (1/n)
n√a = a1n
Example:
2√9 = 912 = 3
You might like to read about Fractional Exponents to discover why!
Summary of Key Properties
| Property | Formula |
|---|---|
| Multiplication | n√ab = n√a × n√b if n is even, then a and b must be ≥ 0 |
| Division | n√a/b = n√a / n√b b can't be zero |
| Fractional Exponents |
n√am = amn the root goes in the bottom of the fraction |
| Important! | n√a + b ≠ n√a + n√b we can't split roots across addition |