How do I fix this surd inside a surd?

[lioric]

Homework Statement

The question says to use the answer to a and b

Relevant Equations

I know I can rationalize. And a surd multiplied by itself removes the root

It’s the last one that I couldn’t do. I have tried the other two. And that’s ok.
You can see the attemp in the picture. I didn’t write it here.
Please help

 

[andrewkirk]

In your working you start with the "=54" there.
That's never going to work because that's essentially what you have to prove (since 54 is the square of $3\sqrt 6$), so you can't use it.

Instead, define $x = 3\sqrt{2 - \sqrt 3}+ \frac 3{ \sqrt{2 - \sqrt 3}}$ and then try to prove that $x^2=54$.
In the first step you can use (a) to simplify $x^2$.
In the second step you can use (b) to simplify further.
You will have some cancellation and collecting of like terms to do.
You should be able to end up with 54.

 

[lioric]

In your working you start with the "=54" there.
That's never going to work because that's essentially what you have to prove (since 54 is the square of $3\sqrt 6$), so you can't use it.

Instead, define $x = 3\sqrt{2 - \sqrt 3}+ \frac 3{ \sqrt{2 - \sqrt 3}}$ and then try to prove that $x^2=54$.
In the first step you can use (a) to simplify $x^2$.
In the second step you can use (b) to simplify further.
You will have some cancellation and collecting of like terms to do.
You should be able to end up with 54.

Just figured it out. Forgot that 54 becomes 3root6

 

[lioric]

Thank you all