How do you rationalize the denominator and simplify `(3sqrt9)/sqrt27`?

You can simplify the expression quite a lot before rationalizing:

• `sqrt(9)=sqrt(3^2)=3`
• `sqrt(27)=sqrt(9*3)=sqrt(9)*sqrt(3)=3sqrt(3)`

So, the expression becomes

`\frac{cancel(3)*3}{cancel(3)sqrt(3)}`

To rationalize this expression, multiply and divide by `sqrt(3)`:

`\frac{3}{sqrt(3)} = \frac{3}{sqrt(3)} * \frac{sqrt(3)}{sqrt(3)} = \frac{3sqrt(3)}{sqrt(3)sqrt(3)} = \frac{cancel(3)sqrt(3)}{cancel(3)} = sqrt(3)`

`(3sqrt9)/sqrt27=(3*3)/(3sqrt3)=3/sqrt3=(3sqrt3)/3=sqrt3`

`(3 sqrt 9)/sqrt 27`

`:.=(3 sqrt 9)/sqrt 27 xx (sqrt 27)/ (sqrt 27)`

`sqrt27*sqrt27=27`

`:.=(3 sqrt(9*27))/27`

`:.=(3 sqrt((3*3)(3*3*3)))/27`

`sqrt3*sqrt 3=3`

`:.=(3*3*3 sqrt 3)/27`

`:.=(cancel27^1 sqrt 3)/cancel27^1`

`:.=sqrt 3`

First, we can rewrite `color(blue)(sqrt27)` as `sqrt3*sqrt9`, which is the same as `3sqrt3`. Doing this leaves us with

`(3sqrt9)/(color(blue)(3sqrt3))`

A `3` in the numerator and denominator cancels, leaving us with

`sqrt9/sqrt3`, which is also equal to

`3/sqrt3`

The key realization is that when we rationalize the denominator, we multiply the top and bottom by that denominator.

In our example, we have

`(3*sqrt3)/(sqrt3*sqrt3)`

Since we are multiplying by `1` essentially, we are not changing the expression's value. The denominator simplifies to `3`, and we're left with

`sqrt3`

Even though we could have rationalized the denominator in the intermediate steps, it never hurts to simplify first.

Hope this helps!