The nth Root

2 √a × √a = a The square root provided two times in a multiplication offers the initial value.

You are watching: Simplify square root of 3 multiplied by the fifth root of 3.

3 3√a × 3√a × 3√a = a The cube root used three times in a multiplication offers the original value.
n n√a × n√a × ... × n√a = a(n of them) The nth root supplied n time in a multiplication gives the initial value.

The nth source Symbol


This is the unique symbol that way "nth root", that is the "radical" prize (used because that square roots) v a small n to average nth root.

Using it

We might use the nth root in a question favor this:

Question: What is "n" in this equation?

n√625 = 5

Answer: i just take place to know that 625 = 54 , so the 4th source of 625 must be 5:

4√625 = 5

Why "Root" ... ?


When you view "root" think

"I know the tree, but what is the source that developed it? "

Example: in √9 = 3 the "tree" is 9 , and the root is 3 .


Now we know what an nth root is, let united state look at some properties:

Multiplication and Division

We deserve to "pull apart" multiplications under the root sign favor this:

n√ab = n√a × n√b (Note: if n is even then a and also b have to both it is in ≥ 0)

This can assist us leveling equations in algebra, and also make part calculations easier:


3√128 = 3√64×2 = 3√64 × 3√2 = 43√2

so the cube root of 128 simplifies come 4 time the cube root of 2.

It likewise works because that division:

n√a/b = n√a / n√b (a≥0 and b>0)Note the b cannot be zero, together we can"t divide by zero

Addition and Subtraction

But us cannot carry out that kind of thing for enhancements 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 favor this:

c = √a2 + b2

Which is not the exact same as c = a + b , right?

It is basic trap to loss into, for this reason beware.

It also method that, unfortunately, additions and subtractions deserve to be hard to resolve when under a source sign.

Exponents vs Roots

An exponent top top one next of "=" have the right to be turned into a source on the other side the "=":

If an = b climate a = n√b

Note: once n is even then b have to be ≥ 0

nth source of a-to-the-nth-Power

When a value has an exponent the n and also we take the nth root us get the value back again ...

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... As soon as a is positive (or zero):

(when a ≥ 0 )



... Or as soon as the exponent is odd :

(when n is odd )



... However when a is negative and the exponent is even we obtain this:


Did you watch that −3 ended up being +3 ?

... So we should do this: (when a

The |a| method the absolute value of a, in various other words any an unfavorable becomes a positive.



So the is something to be mindful of! Read an ext at index number of negative Numbers

Here it is in a little table:

n is weird n is also a ≥ 0 a

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

What happens when the exponent and root are various values (m and also n)?

Well, us are permitted to readjust the order choose this:

n√am = (n√a )m

So this: nth source of (a come the strength m)becomes (nth source of a) come the power m

But over there is one even an ext powerful method ... Us can incorporate the exponent and also root to make a brand-new exponent, choose this: