# How do you simplify #sqrt(108)# ?

##### 2 Answers

Well, first, let's explain how you simplify a square root:

Let's say the problem is this

Our first task is to expand all our components

Now we need to look at our *index*, the

Whatever that number is, that's the value of the exponent we're trying to create. If it's a

So, we are looking for

Back to our example:

Let's condense everything we can to give us a cube root

When we have a number that matches the index, we can take it out of the root:

We can simplify that to

Now, let's solve your problem:

*expand*

*what's the index?*

What can we change to *something* squared

*Bring out the squared*

That gives us

Let's look at the other option

the "simplest" being

#### Explanation:

When

#sqrt(ab) = sqrt(a)sqrt(b)#

Also, if

#sqrt(a^2) = a#

The prime factorisation of

#108 = 2*2*3*3*3#

So we find:

#sqrt(108) = sqrt(2^2*27) = sqrt(2^2)sqrt(27) = 2sqrt(27)#

#sqrt(108) = sqrt(3^2*12) = sqrt(3^2)sqrt(12) = 3sqrt(12)#

#sqrt(108) = sqrt(6^2*3) = sqrt(6^2)sqrt(3) = 6sqrt(3)#

So there are several "simplifications" of

When asked to simplify such a square root, what is normally expected is the expression that no longer has any square factors in the radicand. So in our example,

So if your teacher marked